Splitting of Clifford groups associated to finite abelian groups

SPLITTING OF CLIFF ORD GR OUPS ASSOCIA TED TO FINITE ABELIAN GR OUPS CÉSAR GALINDO Abstract. The Clifford group asso ciated with a finite ab elian group gives rise to a natural extension b y the corresp onding symplectic group. W e prov e that this extension splits as a semidirect pro duct if and only if the group order is not divisible by four. This confirms a conjecture of Korbelář and T olar and extends their cyclic result to arbitrary finite abelian groups. 1. Introduction The Clifford group asso ciated with a finite ab elian group arises naturally in quan tum information theory and in the study of finite Heisenberg groups. In this pap er, C ( A ) denotes the pr oje ctive Clifford group, namely the normalizer of the P auli (or Heisen berg) group mo dulo global phases. In the qubit case, Clifford operators are closely related to the Gottesman–Knill theorem [ 6 , 7 , 1 ]. Our concern here is the extension-theoretic structure of C ( A ) for general finite ab elian groups. While the literature often focuses on systems of n qubits, corresp onding to the abelian group Z n 2 , or on a single qudit of dimension N , corresp onding to Z N , the construction extends naturally to an y finite ab elian group A . F or the qudit setting in arbitrary dimension and the corresp onding Clifford action, see also [ 10 ]. In that setting one sets V A = A ⊕ b A, where b A is the P on try agin dual of A , and one forms the Heisen berg group H ( A ) as a cen tral extension of V A b y the circle group. The Clifford group C ( A ) is then the normalizer of H ( A ) mo dulo global phases. This includes the qubit and qudit cases as sp ecial instances and places the problem in a uniform group-theoretic setting. The Clifford group fits into a natural exact sequence 1 − → V A − → C ( A ) − → Sp( V A ) − → 1 , (1) where Sp ( V A ) is the symplectic group of V A . This extension records the action of Clifford op erators on V A together with the accompan ying phase data. The question is whether this extension splits as a semidirect pro duct. Equiv alently , one asks when the asso ciated cohomology class in H 2 ( Sp ( V A ) , V A ) v anishes. In the cyclic case A = Z N , the o dd and ev en cases are markedly different. F or o dd N , the pro jectiv e Clifford group is known to admit a semidirect-pro duct description ov er the corresp onding symplectic group [ 2 , 9 ]. F or ev en N , K orb elář and T olar [ 11 ] prov ed that the extension fails to split if and only if 4 | N , using a presentation of SL (2 , Z N ) b y generators and relations to derive incompatible constraints on an y hypothetical splitting homomorphism. Motiv ated b y that result, and by its relation to the classical w ork of Bolt, Ro om, an d W all [ 3 , 4 ] on the structure of Clifford collineation groups, they conjectured that for arbitrary finite abelian groups the same divisibility condition should go v ern splitting, namely that the Clifford extension splits if and only if 4 ∤ | A | . The purp ose of this paper is to confirm this conjecture and to determine exactly when the extension ( 1 ) splits. Our main result is the follo wing. 2020 Mathematics Subje ct Classific ation. Primary 20J06, 20C25; Secondary 81P68. K ey wor ds and phr ases. Clifford group, finite Heisenberg group, group extension, group cohomology , W eil represen tation. 1 2 CÉSAR GALINDO Theorem 1.1. L et A b e a finite ab elian gr oup. The Cliffor d extension ( 1 ) splits as a semidir e ct pr o duct if and only if 4 ∤ | A | . The condition 4 ∤ | A | is equiv alen t to requiring that the 2 -primary component A 2 b e either trivial or isomorphic to Z 2 . In particular, Theorem 1.1 recov ers the kno wn o dd-dimensional cyclic splitting and extends it from cyclic groups to arbitrary finite abelian groups of o dd order. More generally , it sho ws that the obstruction to splitting is con trolled en tirely by the 2 -primary part of A , with divisibilit y by four as the threshold. Our pro of differs from that of Korbelář and T olar and accommo dates arbitrary finite ab elian groups. W e first sho w that the splitting problem resp ects primary decomp osition: if A = A 1 ⊕ A 2 with coprime orders, then the extension for A splits if and only if the extensions for A 1 and A 2 split. This reduces the problem to p -primary groups. F or odd primes, we construct explicit splitting sections using square ro ots. The remaining case is the prime 2 , where tw o base cases account for all nonsplitting phenomena. F or the cyclic group A = Z 2 k with k ≥ 2 , we use the pseudo-symplectic mo del to deriv e incompatible lifting constrain ts. F or the elemen tary ab elian group A = Z n 2 with n ≥ 2 , the Clifford group is iden tified with an automorphism group of an extrasp ecial 2 -group, and a classical result of Griess [ 8 ] yields nonsplitting. An em b edding theorem then extends nons plitting from direct summands to the ambien t group. The exceptional case is A = Z 2 , for which Sp ( V A ) ∼ = S 3 and the extension splits. The pap er is organized as follo ws. Section 2 dev elops the Heisen b erg and Clifford groups for finite ab elian groups and introduces the pseudo-symplectic mo del and the obstruction cocycle. Section 3 pro v es coprime functorialit y , constructs an explicit splitting when | A | is o dd, and establishes the reduction to direct summands. Section 4 pro v es nonsplitting for A = Z 2 k . Section 5 treats A = ( Z 2 ) n via extrasp ecial 2 -groups and results of Griess. Section 6 completes the pro of of the splitting criterion. 2. Preliminaries: cohomology and the Clifford extension In this section, w e recall the standard theory of group extensions and cohomology , following the classical treatment b y Brown [ 5 ]. W e then realize the Heisenberg group as a specific cen tral extension asso ciated with V A , and classify suc h extensions using the theory of bicharacters. 2.1. Group extensions and factor sets. Let G and M b e groups, with M ab elian. An extension of G b y M (with M the normal subgroup) is a short exact sequence 1 − → M i − → E π − → G − → 1 . T o describe the group la w of E in terms of G and M , we c ho ose a set-theoretic se ction s : G → E suc h that π ( s ( g )) = g for all g ∈ G , with normalization s (1 G ) = 1 E . This section induces an action of G on M via conjugation: g · m : = s ( g ) i ( m ) s ( g ) − 1 . As M is ab elian, this action is indep endent of the choice of section s , establishing M as a well-defined G -mo dule. In general, the section s is not a homomorphism; consequen tly , the pro duct of sections differs from the section of the product b y a term in M . W e define the factor set or 2-c o cycle β : G × G → M b y the relation: s ( g ) s ( h ) = i ( β ( g , h )) s ( g h ) . Asso ciativit y in E implies the 2-c o cycle identity : ( g · β ( h, k )) β ( g , hk ) = β ( g , h ) β ( g h, k ) . Since the section is normalized, the cocycle is normalized as w ell: β (1 G , g ) = β ( g , 1 G ) = 1 for all g ∈ G . The set of suc h functions forms an abelian group under point wise multiplication, denoted b y Z 2 ( G, M ) . SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 3 Con v ersely , the extension E can b e fully reconstructed from the data ( G, M , · , β ) . The set M × G forms a group under the t wisted multiplication rule: ( m, g ) · ( n, h ) : = ( m ( g · n ) β ( g , h ) , g h ) . Here we implicitly identify M with its image i ( M ) ⊆ E , and the inclusion map no longer app ears in the crossed-pro duct notation. W e denote this crossed pro duct structure by M ⋊ β G . The choice of section s is not unique. If s ′ : G → E is another normalized section, it relates to s via a function λ : G → M (a 1-co c hain) suc h that s ′ ( g ) = i ( λ ( g )) s ( g ) . The corresponding cocycle β ′ satisfies: β ′ ( g , h ) = β ( g , h ) λ ( g )( g · λ ( h )) λ ( g h ) . The term δ ( λ )( g , h ) := λ ( g )( g · λ ( h )) λ ( g h ) is a 2-c ob oundary . The set of all such cob oundaries forms a subgroup B 2 ( G, M ) ⊂ Z 2 ( G, M ) . The quotient group is the second cohomology group of G with co efficien ts in the G -mo dule M : H 2 ( G, M ) : = Z 2 ( G, M ) /B 2 ( G, M ) . The extension E splits (i.e., is isomorphic to the semidirect product M ⋊ G ) if and only if there exists a section s that is a group homomorphism. This occurs precisely when the asso ciated co cycle β is a cob oundary , or equiv alently , when the cohomology class [ β ] v anishes in H 2 ( G, M ) . 2.2. Sc h ur Multiplier of finite ab elian groups. W e sp ecialize the general theory to central extensions of a finite ab elian group V b y the circle group U (1) . F or a finite group V with trivial action on co efficients, the group H 2 ( V , U (1)) ∼ = H 2 ( V , C × ) is the usual Sc h ur multiplier of V (equiv alently , the P on try agin dual of H 2 ( V , Z ) ). In this ab elian setting, these cohomology classes can b e concretely represen ted by bilinear forms. Let Bil ( V ) denote the group of bichar acters , defined as functions B : V × V → U (1) that are homomorphisms in eac h argumen t. Every bic haracter satisfies the 2-cocycle identit y , yielding a natural inclusion Bil( V ) ⊂ Z 2 ( V , U (1)) . W e distinguish t w o subgroups within Bil( V ) : Sym( V ) : = { B ∈ Bil( V ) | B ( u, v ) = B ( v , u ) } , Alt( V ) : = { B ∈ Bil( V ) | B ( u, u ) = 1 } . Note that the alternating condition B ( u, u ) = 1 implies sk ew-symmetry , B ( u, v ) = B ( v , u ) − 1 . The link b et w een co cycles and alternating forms is pro vided by the comm utator. W e define the antisymmetrization map A : Z 2 ( V , U (1)) → Alt( V ) by: A ( β )( u, v ) : = β ( u, v ) β ( v , u ) − 1 . This map measures the non-commutativit y of the extension defined b y β . Restricting A to bic haracters yields a sequence 1 − → Sym( V ) − → Bil( V ) A − − → Alt( V ) − → 1 , (2) whose kernel is precisely the subgroup of symmetric bic haracters. A well-kno wn result (see [ 13 , Proposition 2.1]) establishes that sequence ( 2 ) is exact and we obtain the group isomorphisms Bil( V ) Sym( V ) ∼ = H 2 ( V , U (1)) ∼ − → Alt( V ) . (3) These isomorphisms imply a one-to-one corresp ondence b et w een a cohomology class [ β ] in the Sch ur m ultiplier and the alternating bic haracter ω = A ( β ) . Moreo ver , ev ery 2 -co cycle is cohomologous to a bicharacter. 4 CÉSAR GALINDO 2.3. The Heisenberg extension. Let A b e a finite ab elian group, and let b A = Hom ( A, U (1)) denote its P on try agin dual. The double of A is the direct sum V A : = A ⊕ b A. W e consider the Hilb ert space H = C [ A ] with orthonormal basis {| a ⟩ : a ∈ A } . The fundamental op erators acting on this space are the shift op er ators X a and phase op er ators Z χ , defined for a ∈ A and χ ∈ b A by: X a | b ⟩ : = | a + b ⟩ , Z χ | b ⟩ : = χ ( b ) | b ⟩ . These operators satisfy the comm utation relation Z χ X a = χ ( a ) X a Z χ . F or elemen ts u = ( a u , χ u ) and v = ( a v , χ v ) in V A , we define the W eyl op er ator W u : = X a u Z χ u . The Heisenb er g gr oup H ( A ) is the subgroup of unitary operators generated by these W eyl op erators together with the scalar subgroup U (1) · I : H ( A ) : = { z W u | z ∈ U (1) , u ∈ V A } . A direct computation giv es W u W v = χ u ( a v ) W u + v = β A ( u, v ) W u + v , where the 2 -cocycle β A ∈ Bil( V A ) is giv en b y β A (( a, χ ) , ( b, ψ )) : = χ ( b ) . Th us H ( A ) is a cen tral extension of V A b y U (1) . By Eq. ( 3 ) , the class [ β A ] is determined b y its an tisymmetrization, namely the canonical symple ctic form ω A ( u, v ) : = A ( β A )( u, v ) = χ u ( a v ) χ v ( a u ) − 1 . Consequen tly , W u W v = ω A ( u, v ) W v W u . Since ω A is non-degenerate, the cen ter of H ( A ) is exactly the scalar subgroup U (1) · I , and V A b ecomes a symplectic ab elian group. 2.4. The pseudo-symplectic group Ps ( A ) and the Clifford group. The Cliffor d gr oup C ( A ) is the projective normalizer of the Heisen b erg group, obtained by quotien ting the usual normalizer in U ( H ) b y global phases. Iden tifying U (1) with the scalar subgroup U (1) · I , we set C ( A ) : = N U ( H ) ( H ( A )) U (1) · I =  [ U ] ∈ U ( H ) U (1)     U H ( A ) U † = H ( A )  . Let [ U ] ∈ C ( A ) . Since conjugation by U preserv es H ( A ) and fixes the center U (1) · I p oin twise, it induces an automorphism of the quotien t H ( A ) / ( U (1) · I ) ∼ = V A . Hence there exists a group automorphism T : V A → V A suc h that the W eyl operators transform as U W u U † = λ ( u ) W T u , ∀ u ∈ V A , (4) for some function λ : V A → U (1) . Because conjugation preserv es the m ultiplication la w in H ( A ) , it also preserves the W eyl comm u- tation relation W u W v = ω A ( u, v ) W v W u . Applying conjugation b y U to this iden tit y gives U W u W v U † = ω A ( u, v ) U W v W u U † . Using ( 4 ), w e obtain λ ( u ) λ ( v ) W T u W T v = ω A ( u, v ) λ ( v ) λ ( u ) W T v W T u , SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 5 and therefore W T u W T v = ω A ( u, v ) W T v W T u . Since also W T u W T v = ω A ( T u, T v ) W T v W T u , we conclude that T m ust preserv e the symplectic form: ω A ( T u, T v ) = ω A ( u, v ) . Th us T ∈ Sp( V A ) . Moreov er, comparing the co efficien ts of W T ( u + v ) in U ( W u W v ) U † = ( U W u U † )( U W v U † ) , and using W T u W T v = β A ( T u, T v ) W T ( u + v ) , we obtain λ ( u + v ) λ ( u ) λ ( v ) = β A ( T u, T v ) β A ( u, v ) . (5) F ollowing W eil [ 14 ], w e enco de this data in the pseudo-symple ctic gr oup Ps( A ) . Definition 2.1. The group Ps( A ) is defined as the set of pairs: Ps( A ) : =  ( T , λ ) ∈ Sp( V A ) × Map( V A , U (1))     λ ( u + v ) λ ( u ) λ ( v ) = β A ( T u, T v ) β A ( u, v )  . The group operation is the twisted pro duct (cf. [ 14 , Eq. 7]): ( T , λ ) · ( S, µ ) = ( T S, λ S · µ ) , (6) where ( λ S · µ )( u ) : = λ ( S u ) µ ( u ) , and the identit y elemen t is (id , 1 ) , with 1 the constan t function 1 . Theorem 2.2. The map Φ : Ps ( A ) → C ( A ) , assigning to ( T , λ ) the unique pr oje ctive class of unitary op er ators implementing the action U W u U † = λ ( u ) W T u , is a gr oup isomorphism. Pr o of. W el l-define dness: Let ( T , λ ) ∈ Ps ( A ) . W e will construct a represen tation of H ( A ) with the same cen tral character as the standard one and then apply the Stone-von Neumann-Mack ey theorem. Define ρ ′ : H ( A ) → U ( H ) b y ρ ′ ( z W u ) : = z λ ( u ) W T u . W e v erify that ρ ′ is a represen tation. Using the W eyl m ultiplication la w, we compute ρ ′ ( W u ) ρ ′ ( W v ) = λ ( u ) λ ( v ) W T u W T v = λ ( u ) λ ( v ) β A ( T u, T v ) W T ( u + v ) . Using ( 5 ), this b ecomes ρ ′ ( W u ) ρ ′ ( W v ) = β A ( u, v ) ρ ′ ( W u + v ) = ρ ′ ( β A ( u, v ) W u + v ) = ρ ′ ( W u W v ) . Hence ρ ′ is a unitary represen tation of H ( A ) on H . Ev aluating ( 5 ) at u = v = 0 giv es λ (0) = 1 , and therefore ρ ′ ( z I ) = z λ (0) W T 0 = z I . Th us ρ ′ has the same cen tral character as the standard representation. By the Stone-v on Neumann-Mack ey theorem [ 12 ], the irreducible unitary represen tation of H ( A ) with a fixed central c haracter is unique up to unitary equiv alence. Therefore, there exists an in tert winer U , unique up to a scalar, such that U ρ ( h ) U † = ρ ′ ( h ) , h ∈ H ( A ) , where ρ is the standard represen tation of H ( A ) on H . Restricting to W eyl op erators yields U W u U † = λ ( u ) W T u . 6 CÉSAR GALINDO Homomorphism pr op erty: Let U = Φ( T , λ ) and V = Φ( S, µ ) . The conjugation by the pro duct U V is: ( U V ) W u ( U V ) † = U ( V W u V † ) U † = U ( µ ( u ) W S u ) U † = µ ( u ) λ ( S u ) W T ( S u ) . The resulting phase is µ ( u ) λ ( S u ) = ( λ S · µ )( u ) . The op erator U V th us induces the symplectic map T S and the phase λ S · µ . This matches precisely the pair product defined in Eq. ( 6 ) . Surjectivity follo ws from the discussion preceding the theorem, and injectivit y is immediate from the fact that the pair ( T , λ ) is determined b y the conjugation action U W u U † = λ ( u ) W T u . □ 2.5. The exact sequence and the obstruction co cycle. Via Theorem 2.2 , w e identify C ( A ) with Ps ( A ) and write its elemen ts as pairs ( T , λ ) . Theorem 2.2 realizes the Clifford group as an extension of Sp( V A ) by V A . The projection on to the symplectic comp onen t, π : C ( A ) → Sp( V A ) , π ( T , λ ) = T , is therefore the starting p oint for the extension-theoretic description. The kernel K = k er ( π ) consists of pairs ( id , µ ) with trivial symplectic part. Substituting T = id in to Eq. ( 5 ), w e obtain µ ( u + v ) = µ ( u ) µ ( v ) , hence K iden tifies naturally with the P on try agin dual c V A . Since ω A is non-degenerate, it induces an isomorphism κ A : V A ∼ − → c V A , defined by κ A ( v )( u ) : = ω A ( v , u ) . Using κ A , we iden tify K with V A . The inclusion of the k ernel is then ν : V A → C ( A ) , ν ( v ) = (id , κ A ( v )) . This gives the short exact sequence 1 − → V A ν − → C ( A ) π − → Sp( V A ) − → 1 . (7) This extension induces an action of Sp ( V A ) on V A b y conjugation. If v ∈ V A and g = ( T , λ ) ∈ C ( A ) is any lift of T , then ( T , λ ) ν ( v ) ( T , λ ) − 1 = ν ( T v ) . Hence the induced Sp( V A ) -mo dule structure on V A is the natural one. W e no w construct the 2 -cocycle that represents this extension. Cho ose a set-theoretic section s : Sp( V A ) → C ( A ) , s ( T ) = ( T , λ T ) , where each λ T : V A → U (1) satisfies Eq. ( 5 ). F or T , S ∈ Sp ( V A ) , the elemen t s ( T ) s ( S ) s ( T S ) − 1 lies in k er( π ) = ν ( V A ) , and there is a unique elemen t O s ( T , S ) ∈ V A suc h that s ( T ) s ( S ) s ( T S ) − 1 = ν ( O s ( T , S )) . W riting this k ernel element as a c haracter, we obtain ν ( O s ( T , S )) = (id , Γ T ,S ) , Γ T ,S = λ S T · λ S λ T S . Since Γ T ,S is a linear c haracter of V A , the defining relation for O s ( T , S ) is ω A ( O s ( T , S ) , u ) = λ T ( S u ) λ S ( u ) λ T S ( u ) . (8) SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 7 The map O s : Sp( V A ) × Sp( V A ) → V A is the obstruction 2 -c o cycle associated with the section s . Its cohomology class [ O s ] ∈ H 2 (Sp( V A ) , V A ) is indep endent of the c hoice of section. W e denote this intrinsic extension class b y [ O A ] : = [ O s ] ∈ H 2 (Sp( V A ) , V A ) . It is the obstruction to splitting the exact sequence ( 7 ). 3. Decomposition and splitting reductions In this section, we reduce the splitting problem to the 2 -primary case. W e first show that the obstruction class decomp oses along coprime factors, then prov e that it v anishes for groups of o dd order, and finally sho w that splitting descends to direct summands. T ogether, these results reduce the pro of of the splitting criterion to explicit 2 -primary base cases. 3.1. Coprime decomp osition. W e first address the general case of a decomposition in to coprime factors. Let A = A 1 ⊕ A 2 , gcd( | A 1 | , | A 2 | ) = 1 , where A 1 and A 2 are finite abelian groups. This decomposition extends naturally to the double, V A = V A 1 ⊕ V A 2 , and the symplectic form splits as the orthogonal sum ω A = ω A 1 ⊕ ω A 2 . The symplectic group respects this splitting. Let T ∈ Sp ( V A ) . Because automorphisms preserv e elemen t orders, the subsets of elements whose orders divide | V A 1 | and | V A 2 | are characteristic in V A . Since | V A 1 | and | V A 2 | are coprime, these subsets are precisely the direct summands V A 1 and V A 2 . Hence b oth summands are T -in v ariant, and every symplectic automorphism decomp oses uniquely . W e obtain the isomorphism Sp( V A ) ∼ = Sp( V A 1 ) × Sp( V A 2 ) , ( T 1 , T 2 ) 7→ T 1 ⊕ T 2 . Via Theorem 2.2 , we ma y describ e elemen ts of eac h Clifford group as pairs ( T i , λ i ) satisfying the cob oundary condition. W e define Ψ : C ( A 1 ) × C ( A 2 ) → C ( A 1 ⊕ A 2 ) (( T 1 , λ 1 ) , ( T 2 , λ 2 )) 7→ ( T 1 ⊕ T 2 , λ 1 ⊕ λ 2 ) , where ( λ 1 ⊕ λ 2 )( u 1 + u 2 ) : = λ 1 ( u 1 ) λ 2 ( u 2 ) , u i ∈ V A i . Lik ewise, the bic haracter on V A = V A 1 ⊕ V A 2 is given b y β A ( u 1 + u 2 , v 1 + v 2 ) = β A 1 ( u 1 , v 1 ) β A 2 ( u 2 , v 2 ) . A direct v erification sho ws that λ 1 ⊕ λ 2 satisfies the coboundary condition for β A . Prop osition 3.1. L et A = A 1 ⊕ A 2 with gcd ( | A 1 | , | A 2 | ) = 1 . The map Ψ induc es an isomorphism of short exact se quenc es: 1 V A 1 ⊕ V A 2 C ( A 1 ) × C ( A 2 ) Sp( V A 1 ) × Sp( V A 2 ) 1 1 V A C ( A ) Sp( V A ) 1 ∼ = Ψ ∼ = ∼ = 8 CÉSAR GALINDO Conse quently, the obstruction class de c omp oses: [ O A ] = [ O A 1 ] ⊕ [ O A 2 ] ∈ H 2 (Sp( V A ) , V A ) . Pr o of. The commutativit y of the diagram follo ws from the definition of Ψ . On the k ernel, Ψ maps the pair of phases ( κ A 1 ( v 1 ) , κ A 2 ( v 2 )) to the pro duct character κ A 1 ( v 1 ) κ A 2 ( v 2 ) = κ A ( v 1 + v 2 ) , whic h is exactly the canonical iden tification V A 1 ⊕ V A 2 ∼ = V A . On the quotient, Ψ induces the isomorphism Sp( V A 1 ) × Sp( V A 2 ) ∼ = Sp( V A ) describ ed ab ov e. The Fiv e Lemma therefore implies that Ψ is an isomorphism. It follo ws that the extension class for C ( A ) is the image of the pro duct extension class, which corresp onds to the sum of the t w o obstruction classes. □ W e apply this result to the primary decomp osition of A . W e can uniquely decomp ose the group as: A = A odd ⊕ A 2 , where A odd con tains all elemen ts of odd order and A 2 is the 2 -primary comp onent of A . Since | A 2 | and | A odd | are coprime, Prop osition 3.1 implies that the splitting problem for C ( A ) reduces to the indep enden t analysis of C ( A odd ) and C ( A 2 ) . 3.2. Splitting at o dd primes. W e no w prov e that the obstruction v anishes whenever | A | is o dd b y constructing an explicit splitting of the Clifford extension. In this setting, the v alues of an y bic haracter lie in the subgroup of ro ots of unit y of o dd order. The squaring map z 7→ z 2 is therefore an automorphism of this subgroup, and square ro ots are w ell defined and unique. W e denote the unique square root of an element z by z 1 / 2 . Recall from ( 2 ) that the an tisymmetrization map A : Bil( V A ) → Alt( V A ) is surjective. If w e restrict this map to the subgroup of alternating bic haracters Alt ( V A ) ⊂ Bil ( V A ) , then A ( B )( u, v ) = B ( u, v ) · B ( u, v ) = B ( u, v ) 2 . Since squaring is an automorphism on ro ots of unit y of odd order, the restriction A| Alt( V A ) is an isomorphism. Th us ev ery alternating form ω ∈ Alt ( V A ) admits a unique alternating square root ω 1 / 2 ∈ Alt( V A ) such that A ( ω 1 / 2 ) = ω . This allows us to define a candidate for a splitting. W e no w construct the function λ T : V A → U (1) needed for the Clifford action as λ T ( u ) : =  β A ( T u, T u ) β A ( u, u )  1 / 2 . (9) T o sho w that this defines a section, w e first v erify the coboundary condition ( 5 ) . Using the expansion β A ( u + v , u + v ) = β A ( u, u ) β A ( v , v ) β A ( u, v ) β A ( v , u ) and the fact that T preserv es ω A = A ( β A ) , we compute the square of the ratio:  λ T ( u + v ) λ T ( u ) λ T ( v )  2 = β A ( T ( u + v ) , T ( u + v )) β A ( u + v , u + v ) β A ( u, u ) β A ( T u, T u ) β A ( v , v ) β A ( T v , T v ) = β A ( T u, T v ) β A ( T v , T u ) β A ( u, v ) β A ( v , u ) =  β A ( T u, T v ) β A ( u, v )  2 . SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 9 Since the squaring map is injectiv e on ro ots of unit y of o dd order, the desired equalit y follows. Hence s : T 7→ ( T , λ T ) defines a section Sp ( V A ) → Ps ( A ) , and therefore, via Theorem 2.2 , a section into C ( A ) . It remains to pro ve that this section is a homomorphism. Equiv alently , w e must show that the asso ciated obstruction co cycle v anishes. The co cycle O s ( T , S ) measures the failure of the section s to b e a homomorphism; see ( 8 ) . Squaring that iden tity and substituting λ T ( u ) 2 = β A ( T u, T u ) β A ( u, u ) − 1 , we obtain ω A ( O s ( T , S ) , u ) 2 = λ T ( S u ) 2 λ S ( u ) 2 λ T S ( u ) 2 =  β A ( T S u,T S u ) β A ( S u,S u )   β A ( S u,S u ) β A ( u,u )   β A ( T S u,T S u ) β A ( u,u )  = 1 . Since elements of V A ha v e odd order, every v alue of ω A also has o dd order, and the equation x 2 = 1 therefore implies x = 1 . Hence ω A ( O s ( T , S ) , u ) = 1 for all u . By the non-degeneracy of ω A , this forces O s ( T , S ) = 0 . Prop osition 3.2. If | A | is o dd, the obstruction class [ O A ] vanishes. The Cliffor d gr oup splits as a semidir e ct pr o duct C ( A ) ∼ = V A ⋊ Sp ( V A ) , with the splitting given explicitly by the se ction λ T in Eq. ( 9 ) . Com bining this result with Prop osition 3.1 , we conclude that for an y finite ab elian group, the non-trivial cohomological structure resides entirely in the 2-primary comp onent. 3.3. Reduction to direct summands. W e no w sho w that splitting descends to direct summands. Let A = B ⊕ C b e an arbitrary decomp osition of finite ab elian groups; no coprimalit y assumption is imp osed. Then V A decomp oses orthogonally as V A = V B ⊕ V C . There are natural em b eddings i : V B  → V A , i ( v B ) = ( v B , 0) , and j : Sp( V B )  → Sp( V A ) , j ( S ) = S ⊕ id V C . Using the pseudo-symplectic description from Theorem 2.2 , we can lift j to the Clifford level. F or ( S, µ ) ∈ Ps( B ) , define ι ( S, µ ) : = ( S ⊕ id V C , µ · 1 ) , where ( µ · 1 )( v B , v C ) : = µ ( v B ) . Since β A = β B · β C , the function µ · 1 satisfies the coboundary condition for β A ; the C -comp onen t is trivial because the symplectic map acts as the iden tit y on V C . Moreo v er, the t wisted pro duct is preserv ed, and therefore ι is a group homomorphism. This construction yields a commutativ e diagram of exact sequences (identifying C ( A ) ∼ = Ps( A ) ): 1 V B C ( B ) Sp( V B ) 1 1 V A C ( A ) Sp( V A ) 1 i ι j Theorem 3.3. L et A = B ⊕ C b e a de c omp osition of finite ab elian gr oups. If the Cliffor d extension for A splits, then the Cliffor d extension for B splits. 10 CÉSAR GALINDO Pr o of. Assume that the extension for A splits. Then there exists a normalized group homomorphism σ A : Sp( V A ) → C ( A ) satisfying π A ◦ σ A = id Sp( V A ) . W e construct from it a splitting section for the Clifford extension for B . F or S ∈ Sp( V B ) , set Σ S = σ A ( j ( S )) ∈ C ( A ) . Via the iden tification C ( A ) ∼ = Ps( A ) from Theorem 2.2 , we ma y write Σ S ∼ = ( S ⊕ id V C , Λ S ) , where Λ S : V A → U (1) satisfies the cob oundary condition Λ S ( u + v ) Λ S ( u )Λ S ( v ) = β A (( S ⊕ id) u, ( S ⊕ id) v ) β A ( u, v ) . W e no w restrict Λ S to the cop y of V B inside V A . Define λ S ( u B ) : = Λ S ( u B , 0 C ) . W e verify that λ S satisfies the cob oundary condition for β B . F or u = ( u B , 0) and v = ( v B , 0) in V B ⊕ { 0 } , the factorization β A = β B · β C giv es β A ( u, v ) = β B ( u B , v B ) · β C (0 , 0) = β B ( u B , v B ) , since β C (0 , 0) = 1 . Similarly , ( S ⊕ id)( u B , 0) = ( S u B , 0) , and therefore λ S ( u B + v B ) λ S ( u B ) λ S ( v B ) = β B ( S u B , S v B ) β B ( u B , v B ) . This confirms that ( S, λ S ) is a v alid elemen t of Ps( B ) . Define σ B : Sp( V B ) → Ps( B ) , σ B ( S ) : = ( S, λ S ) . It remains to pro v e that σ B is a group homomorphism. Let S 1 , S 2 ∈ Sp( V B ) . Since σ A is a homomorphism, Σ S 1 S 2 = Σ S 1 · Σ S 2 . Using the t wisted pro duct in Ps( A ) , we obtain Σ S 1 · Σ S 2 = ( S 1 S 2 ⊕ id V C , Λ S 2 ⊕ id S 1 · Λ S 2 ) , where (Λ S 2 ⊕ id S 1 · Λ S 2 )( u ) = Λ S 1 (( S 2 ⊕ id) u ) · Λ S 2 ( u ) . Comparing with Σ S 1 S 2 = ( S 1 S 2 ⊕ id V C , Λ S 1 S 2 ) , we obtain Λ S 1 S 2 ( u ) = Λ S 1 (( S 2 ⊕ id) u ) · Λ S 2 ( u ) . Restricting to u = ( u B , 0) ∈ V B ⊕ { 0 } gives λ S 1 S 2 ( u B ) = Λ S 1 S 2 ( u B , 0) = Λ S 1 (( S 2 ⊕ id)( u B , 0)) · Λ S 2 ( u B , 0) = Λ S 1 ( S 2 u B , 0) · Λ S 2 ( u B , 0) = λ S 1 ( S 2 u B ) · λ S 2 ( u B ) . This is precisely the twisted pro duct rule for the phases in Ps( B ) : ( S 1 , λ S 1 ) · ( S 2 , λ S 2 ) = ( S 1 S 2 , λ S 2 S 1 · λ S 2 ) = ( S 1 S 2 , λ S 1 S 2 ) . Hence σ B ( S 1 S 2 ) = σ B ( S 1 ) · σ B ( S 2 ) , and therefore σ B is a group homomorphism. Finally , σ B is normalized b ecause σ B ( id ) = ( id , λ id ) and λ id ( u B ) = Λ id ( u B , 0) = 1 , using that σ A is normalized. Since π B ( σ B ( S )) = S b y construction, σ B is a splitting section for the Clifford extension for B . □ SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 11 4. Non-splitting for Z 2 k In this section, we establish that the Clifford extension do es not split when A = Z N with N = 2 k and k ≥ 2 . This non-splitting result w as first established b y K orb elář and T olar [ 11 ], who prov ed that for ev en N , the Clifford group splits if and only if N is not divisible b y 4 . Their pro of builds on the standard description of the cyclic Clifford group and its symplectic action [ 2 ]. Here, we provide an alternative deriv ation utilizing the pseudo-symplectic description dev eloped in Theorem 2.2 . W e k eep the discussion entirely in the standard modular presen tation generated b y the matrices in ( 10 ) , whic h allows us to reframe the obstruction explicitly in terms of cohomological phase data. This giv es an alternativ e deriv ation of their result from the modular relations, indep endent of the sp ecific matrix representations used in [ 11 ]. F or A = Z N , we iden tify d Z N ∼ = Z N via the standard c haracter pairing and write V Z N : = Z N ⊕ Z N , with elemen ts u = ( a, p ) and v = ( b, q ) . In the computations b elow, whenever s uc h co ordinates occur inside exponentials or parit y expressions, we take their standard representativ es in { 0 , . . . , N − 1 } ⊂ Z . The Heisenberg 2-cocycle is β Z N ( u, v ) = exp  2 π i N pb  , and we use Theorem 2.2 to identify C ( Z N ) with pairs ( T , λ ) satisfying the cob oundary condition λ ( u + v ) λ ( u ) λ ( v ) = β Z N ( T u, T v ) β Z N ( u, v ) . Since V Z N is free of rank 2 ov er Z N with its standard alternating form, we identify Sp ( V Z N ) with Sp(2 , Z N ) = SL(2 , Z N ) . The group SL(2 , Z N ) is generated b y: t =  1 1 0 1  , s =  0 − 1 1 0  . (10) Their action on V Z N is giv en by t ( a, p ) = ( a + p, p ) and s ( a, p ) = ( − p, a ) . A splitting homomorphism σ : SL (2 , Z N ) → C ( Z N ) is determined b y its v alues ˜ t = σ ( t ) and ˜ s = σ ( s ) . The argumen t pro ceeds in tw o steps. W e first classify all lifts of the generators s and t to C ( Z N ) . W e then show that the relations t N = I and ( st ) 3 = s 2 , whic h any homomorphism must preserv e, imp ose incompatible conditions on the parameter x app earing in the lift of t , which rules out the existence of a splitting. 4.1. Explicit lifts of the generators. T o analyze a potential splitting, we determine the admissible lifts of the generators to the Clifford group. These form ulas reduce the problem to the parameters app earing in the phase functions, and the contradiction in the next subsection will ultimately depend only on the parameter x in the lift of t . F or the generator t , we compute the ratio of co cycles directly using the action t ( a, p ) = ( a + p, p ) . F or an y u = ( a, p ) and v = ( b, q ) : β Z N ( tu, tv ) β Z N ( u, v ) = exp( 2 π i N p ( b + q )) exp( 2 π i N pb ) = exp  2 π i N pq  . A particular solution is λ (0) t ( a, p ) = exp( π ip 2 / N ) , since λ (0) t ( u + v ) λ (0) t ( u ) λ (0) t ( v ) = exp  π i N  ( p + q ) 2 − p 2 − q 2   = exp  2 π i N pq  . 12 CÉSAR GALINDO An y other solution differs from λ (0) t b y a linear character of V Z N . Parametrizing c haracters as χ ( x,y ) ( a, p ) = exp(2 π i ( xa + y p ) / N ) , the general lift of t takes the form: ˜ t = ( t, λ t ) , λ t ( a, p ) = exp  π i N p 2 + 2 π i N ( xa + y p )  , (11) determined by a unique pair ( x, y ) ∈ Z 2 N . F or the generator s , the cob oundary ratio is: β Z N ( su, sv ) β Z N ( u, v ) = exp( 2 π i N a ( − q )) exp( 2 π i N pb ) = exp  − 2 π i N ( aq + pb )  . A particular solution is λ (0) s ( a, p ) = exp( − 2 π i ap/ N ) , since λ (0) s ( u + v ) λ (0) s ( u ) λ (0) s ( v ) = exp  − 2 π i N  ( a + b )( p + q ) − ap − bq   = exp  − 2 π i N ( aq + bp )  . Hence every lift of s is of the form ˜ s = ( s, λ s ) , λ s ( a, p ) = exp  − 2 π i N ap + 2 π i N ( z a + w p )  , (12) for a pair ( z , w ) ∈ Z 2 N . 4.2. Incompatible lifting constrain ts. W e now deriv e the con tradiction. Assuming that a splitting exists, the lifted generators must satisfy , in particular, the relations t N = I and ( st ) 3 = s 2 . The relation t N = I forces the parameter x in the lift of t to b e o dd, whereas the relation ( st ) 3 = s 2 forces 2 x ≡ 0 ( mo d N ) . F or N = 2 k with k ≥ 2 , these conditions are incompatible. Note that no additional order condition on the lift of s will b e needed. Lemma 4.1. L et N = 2 k with k ≥ 1 . If ˜ t N = ( I , 1) , then the p ar ameter x in ( 11 ) must b e o dd. Pr o of. Using the pro duct law ( T , λ ) · ( S, µ ) = ( T S, λ S µ ) iteratively , the phase of ˜ t N is given b y: Λ N ( u ) = N − 1 Y j =0 λ t ( t j u ) . Since t j ( a, p ) = ( a + j p, p ) , the second comp onen t p remains constant. Substituting λ t from ( 11 ): Λ N ( a, p ) = N − 1 Y j =0 exp  π i N p 2 + 2 π i N ( x ( a + j p ) + y p )  . W e separate the pro duct in to quadratic and linear parts. The quadratic term is constan t in j : N − 1 Y j =0 exp  π i N p 2  = exp( π ip 2 ) = ( − 1) p 2 = ( − 1) p , where we used that p 2 ≡ p (mo d 2) for integers. The linear term inv olv es a sum o v er j : N − 1 X j =0 2 π i N ( xa + xj p + y p ) = 2 π i ( xa + y p ) + 2 π i N xp N − 1 X j =0 j. Using P N − 1 j =0 j = N ( N − 1) 2 , the second term b ecomes π ixp ( N − 1) . Since N is ev en, N − 1 is o dd; therefore exp( π ixp ( N − 1)) = ( − 1) xp . The term exp(2 π i ( xa + y p )) = 1 . Thus, the total phase is: Λ N ( a, p ) = ( − 1) p ( − 1) xp = ( − 1) p (1+ x ) . F or ˜ t N = ( I , 1) , we require Λ N ( a, p ) = 1 for all p . This holds if and only if 1 + x is even, i.e., x ≡ 1 (mo d 2) . □ SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 13 Lemma 4.2. L et N = 2 k with k ≥ 1 . Cho ose lifts ˜ t = ( t, λ t ) and ˜ s = ( s, λ s ) as in ( 11 ) – ( 12 ) with p ar ameters ( x, y ) and ( z , w ) r esp e ctively. If the mo dular r elation ( ˜ s ˜ t ) 3 = ˜ s 2 holds in the Cliffor d gr oup, then 2 x ≡ 0 (mo d N ) . Pr o of. W e organize the computation in tw o steps: first w e v erify the relation for a conv enient pair of reference lifts, and then we determine ho w the relation c hanges after t wisting by arbitrary linear c haracters. Step 1: The reference lifts satisfy ( ˜ s 0 ˜ t 0 ) 3 = ˜ s 2 0 . Fix the particular solutions λ (0) t ( a, p ) := exp  π i N p 2  , λ (0) s ( a, p ) := exp  − 2 π i N ap  , and set ˜ t 0 := ( t, λ (0) t ) , ˜ s 0 := ( s, λ (0) s ) . First, ˜ s 2 0 = ( s 2 , ( λ (0) s ) s λ (0) s ) = ( − I , 1) , b ecause ( λ (0) s ) s λ (0) s ≡ 1 . Next let ˜ g 0 := ˜ s 0 ˜ t 0 = ( st, λ g 0 ) , where λ g 0 ( a, p ) = ( λ (0) s ) t ( a, p ) λ (0) t ( a, p ) = exp  − 2 π i N ( a + p ) p  exp  π i N p 2  = exp  − 2 π i N ap  exp  − π i N p 2  . Since g := st satisfies g 3 = − I = s 2 in SL(2 , Z N ) , it remains to compare the phase part. Using ( ˜ s 0 ˜ t 0 ) 3 = ˜ g 3 0 = ( g 3 , Λ 0 ) , Λ 0 ( u ) = λ g 0 ( g 2 u ) λ g 0 ( g u ) λ g 0 ( u ) , a direct substitution with g ( a, p ) = ( − p, a + p ) sho ws that Λ 0 ( u ) = 1 for all u . Therefore ( ˜ s 0 ˜ t 0 ) 3 = ( − I , 1) = ˜ s 2 0 . Step 2: Reduction to a linear-c haracter computation. W e now compare arbitrary lifts with the reference lifts from Step 1. The mo dular relation will b e tested by isolating the residual character in the w ord W := ( ˜ s ˜ t ) 3 ˜ s − 2 . If this residual c haracter is trivial, then the mo dular relation holds; otherwise it fails. Let Ξ := { ( I , χ v ) : v ∈ Z 2 N } ≤ C ( Z N ) , where χ v ( u ) := exp  2 π i N ⟨ v , u ⟩  , ⟨ v , u ⟩ := v 1 u 1 + v 2 u 2 (mo d N ) for v = ( v 1 , v 2 ) and u = ( u 1 , u 2 ) in Z 2 N . F or an y S ∈ SL (2 , Z N ) w e hav e χ S v ( u ) := χ v ( S u ) = χ S T v ( u ) , hence ( H , χ v 1 )( K, χ v 2 ) = ( H K, χ K T v 1 + v 2 ) , ( H, χ v ) − 1 = ( H − 1 , χ − H − T v ) . (13) In particular, Ξ is normal in C ( Z N ) and conjugation acts by v 7→ H T v : ( H , λ ) ( I , χ v ) = ( I , χ H T v ) ( H , λ )  ( H , λ ) ∈ C ( Z N )  . W rite the general lifts as ˜ t = ( t, λ (0) t χ v t ) = ˜ t 0 ( I , χ v t ) , ˜ s = ( s, λ (0) s χ v s ) = ˜ s 0 ( I , χ v s ) , where v t =  x y  and v s =  z w  . By Step 1, W 0 := ( ˜ s 0 ˜ t 0 ) 3 ˜ s − 2 0 = ( I , 1) . W riting ξ s := ( I , χ v s ) and ξ t := ( I , χ v t ) , we ha v e ˜ s = ˜ s 0 ξ s and ˜ t = ˜ t 0 ξ t . Using the relation ( H , λ )( I , χ v ) = ( I , χ H T v )( H , λ ) , w e comm ute all c haracter factors to the righ t. The non-c haracter part then collapses to W 0 = ( I , 1) , and therefore W = ( I , χ v W ) for a uniquely determined v W ∈ Z 2 N . 14 CÉSAR GALINDO W e compute v W explicitly . F rom ( 13 ), the c haracter parameter of ˜ s ˜ t is v st = t T v s + v t =  1 0 1 1  z w ! + x y ! = x + z z + w + y ! . Let g := st , hence g T =  0 1 − 1 1  . Then ( g , χ v st ) 3 = ( g 3 , χ (( g T ) 2 + g T + I ) v st ) , ( g T ) 2 + g T + I =  0 2 − 2 2  , and therefore v ( st ) 3 =  0 2 − 2 2  x + z z + w + y ! = 2( z + w + y ) 2( w + y − x ) ! . Next, v s 2 = ( s T + I ) v s =  1 1 − 1 1  z w ! = z + w w − z ! . Since ( st ) 3 = s 2 = − I , the final m ultiplication gives v W = − v ( st ) 3 + v s 2 = − z − w − 2 y − z − w − 2 y + 2 x ! . If the mo dular relation ( ˜ s ˜ t ) 3 = ˜ s 2 holds, then W = ( I , 1) , implying that χ v W is trivial and v W ≡ 0 (mo d N ) . Subtracting the t w o comp onents yields 2 x ≡ 0 (mo d N ) . □ Theorem 4.3. L et N = 2 k with k ≥ 2 . The Cliffor d extension 1 − → V Z N − → C ( Z N ) − → SL(2 , Z N ) − → 1 do es not split. Pr o of. Supp ose the extension splits via a homomorphism σ : SL (2 , Z N ) → C ( Z N ) . Let ˜ t = σ ( t ) and ˜ s = σ ( s ) b e the images of the generators, with ˜ t = ( t, λ t ) parametrized b y ( x, y ) ∈ Z 2 N as in ( 11 ). Since σ is a homomorphism, the lifts m ust satisfy the defining relations of SL (2 , Z N ) . Lemmas 4.1 and 4.2 sho w that these relations imp ose incompatible conditions on the same parameter x . More precisely: • The relation t N = I implies ˜ t N = ( I , 1) , hence b y Lemma 4.1 , x m ust b e o dd. • The relation ( st ) 3 = s 2 implies ( ˜ s ˜ t ) 3 = ˜ s 2 , thus b y Lemma 4.2 , 2 x ≡ 0 (mo d N ) . F or N = 2 k with k ≥ 2 , the condition 2 x ≡ 0 ( mo d N ) implies x ∈ { 0 , N / 2 } . Since N / 2 = 2 k − 1 is ev en for k ≥ 2 , b oth possible v alues of x are even. This contradicts the requirement that x b e o dd. Therefore, no splitting homomorphism exists. □ R emark 4.4 . The argument does not require imp osing any order constrain ts on the lift ˜ s . The con tradiction arises solely from the incompatibilit y b etw een the relations t N = I and ( st ) 3 = s 2 . 5. Symplectic obstruction in the element ar y abelian case In this section w e establish nonsplitting of the Clifford extension for A = ( Z 2 ) n with n ≥ 2 b y relating it to the automorphism extension of an extrasp ecial 2 -group studied by Griess [ 8 ]. W e first describ e the relev ant extension attached to the n -qubit Pauli group and its extrasp ecial model. W e then identify it with the Clifford extension. The desired nonsplitting statemen t then follows from Griess’ results. SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 15 5.1. The P auli group and its automorphisms. Fix A = ( Z 2 ) n with V A = A ⊕ b A ∼ = ( F 2 ) 2 n . The n -qubit Pauli gr oup P n is the finite subgroup of U ( H ) generated b y the W eyl op erators and the scalar i : P n := ⟨ W u , iI | u ∈ V A ⟩ . Explicitly , P n = { z W u | z ∈ ⟨ i ⟩ , u ∈ V A } has order 2 2 n +2 . Its cen ter is Z ( P n ) = ⟨ iI ⟩ ∼ = Z 4 , and the quotien t P n / Z ( P n ) is canonically isomorphic to V A . T o connect this with the classical theory of extrasp ecial 2 -groups, w e view P n as a central pro duct. Let Y := ⟨ iI ⟩ ∼ = Z 4 , and let E n b e the extrasp ecial 2 -group of order 2 2 n +1 generated by the W eyl op erators mo dulo the central in v olution ⟨− I ⟩ ∼ = Z 2 . Then P n ∼ = E n ◦ Y , where the cen tral pro duct is formed by amalgamating the common subgroup ⟨− I ⟩ . Here E n is the extrasp ecial 2 -group of plus t yp e, equiv alen tly a cen tral product of n copies of D 8 . This is the form in which Griess’ results apply . F ollowing Griess [ 8 ], w e consider the subgroup A ut Y ( E n ◦ Y ) := { ϕ ∈ A ut( E n ◦ Y ) | ϕ | Y = id Y } , of automorphisms that fix Y p oin twise. In Griess’ notation, this group is denoted b y A ( E n ◦ Y ) . Ev ery such automorphism induces a well-defined automorphism of the quotien t ( E n ◦ Y ) / Y ∼ = V A . Because ϕ fixes the center, it preserves the comm utator form and hence the induced map on V A lies in Sp ( V A ) . The k ernel consists of inner automorphisms, which iden tify with V A through the symplectic form. Therefore w e obtain a short exact sequence 1 − → V A − → A ut Y ( E n ◦ Y ) ρ − → Sp( V A ) − → 1 , (14) whic h is the automorphism extension to b e compared with the Clifford extension. 5.2. Iden tification with the Clifford extension. W e no w sho w that the automorphism exten- sion ( 14 ) coincides with the Clifford extension. The k ey p oint is that both middle terms are describ ed b y the same pairs ( T , λ ) satisfying the same compatibilit y condition. Let T ∈ Sp ( V A ) . An automorphism ϕ ∈ A ut Y ( E n ◦ Y ) lifting T m ust satisfy ϕ ( W u ) = λ ( u ) W T u for some function λ : V A → ⟨ i ⟩ . Imp osing that ϕ preserv es m ultiplication yields the cob oundary condition λ ( u + v ) λ ( u ) λ ( v ) = β A ( T u, T v ) β A ( u, v ) . (15) This is precisely condition ( 5 ) from Theorem 2.2 , with the phase function λ taking v alues in ⟨ i ⟩ rather than U (1) . Since the righ t-hand side of ( 15 ) lies in ⟨ i ⟩ for elementary ab elian 2 -groups, the t w o conditions coincide. Prop osition 5.1. Ther e is a c anonic al isomorphism of short exact se quenc es 1 V A C ( A ) Sp( V A ) 1 1 V A A ut Y ( E n ◦ Y ) Sp( V A ) 1 π ∼ = ρ identifying b oth midd le terms with p airs ( T , λ ) satisfying ( 15 ) . Pr o of. The discussion abov e defines a map Φ : A ut Y ( E n ◦ Y ) − → C ( A ) , 16 CÉSAR GALINDO sending an automorphism ϕ with ϕ ( W u ) = λ ( u ) W T u to the pair ( T , λ ) ∈ Ps ( A ) ∼ = C ( A ) . Since b oth groups use the same multiplication rule ( T , λ ) · ( S, µ ) = ( T S, λ S µ ) , the map Φ is a group homomorphism. The diagram comm utes by construction. On the quotient, Φ preserves the symplectic comp onent, hence the right square commutes. On the kernel, b oth extensions identify the kernel with V A : on the Clifford side through the c haracter description determined by ω A , and on the automorphism side through inner automorphisms. Under these iden tifications, Φ acts as the iden tity on V A , and the left square comm utes as w ell. Since the left and righ t vertical maps are isomorphisms, the Fiv e Lemma implies that the middle v ertical map is an isomorphism as well. □ 5.3. Nonsplitting for n ≥ 2 . By Prop osition 5.1 , the splitting problem for the Clifford extension is equiv alen t to the corresponding splitting problem for the automorphism extension ( 14 ) . W e can therefore apply Griess’ nonsplitting results directly . Theorem 5.2. L et A = ( Z 2 ) n . The Cliffor d extension 1 − → V A − → C ( A ) − → Sp( V A ) − → 1 splits if and only if n = 1 . Pr o of. F or n = 1 , we ha v e Sp ( V A ) = Sp (( F 2 ) 2 ) ∼ = S 3 and P 1 ∼ = D 8 . In this case the automorphism extension splits: explicitly , A ut Y ( D 8 ◦ Y ) ∼ = S 4 , and the subgroup S 3 ≤ S 4 fixing one p oint is a complemen t to V A ∼ = Z 2 2 . F or n ≥ 2 , Griess’ results show that the automorphism extension is nonsplit. By [ 8 , Corollary 2], the extension 1 − → Inn( E n ◦ Y ) − → Aut Y ( E n ◦ Y ) − → Sp(2 n, F 2 ) − → 1 is nonsplit for n ≥ 3 , and by [ 8 , Corollary 3] the same holds for n = 2 . Prop osition 5.1 transp orts this conclusion to the Clifford extension. □ 6. Proof of the Splitting Criterion W e now combine the reduction to the 2 -primary part with the t w o base nonsplitting results, namely the cyclic 2 -p ow er case and the elementary abelian case, to complete the pro of of the splitting criterion. Theorem 6.1. L et A b e a finite ab elian gr oup and let V A = A ⊕ b A . The Cliffor d extension 1 − → V A − → C ( A ) − → Sp( V A ) − → 1 (16) splits if and only if 4 ∤ | A | . Equivalently, the obstruction class O A ∈ H 2 ( Sp ( V A ) , V A ) vanishes if and only if 4 ∤ | A | . Pr o of. This is the statemen t announced in Theorem 1.1 . W rite the primary decomp osition A ∼ = A odd ⊕ A 2 , where | A odd | is o dd and | A 2 | is a p ow er of 2 . By Prop osition 3.1 , the Clifford extension class decomp oses as [ O A ] = [ O A odd ] ⊕ [ O A 2 ] , and hence ( 16 ) splits if and only if the 2 -primary extension for A 2 splits. Moreo v er, b y Prop osition 3.2 , w e hav e [ O A odd ] = 0 . Thus it suffices to determine when the extension splits for A 2 . Step 1: the splitting c ases. If A 2 = 1 , then A has o dd order and the extension splits by Prop osition 3.2 . If A 2 ∼ = Z 2 , then the extension splits by Theorem 5.2 , whic h treats the exceptional rank-one elemen tary ab elian case. SPLITTING OF CLIFF ORD GROUPS ASSOCIA TED TO FINITE ABELIAN GROUPS 17 Step 2: the nonsplitting c ases. Assume | A 2 | ≥ 4 . W rite A 2 ∼ = Z 2 k 1 ⊕ · · · ⊕ Z 2 k m , k 1 ≥ · · · ≥ k m ≥ 1 . There are then exactly tw o p ossibilities: (A) A 2 has an element of or der ≥ 4 . Equiv alently , k 1 ≥ 2 , hence Z 2 k 1 is a direct summand of A 2 . By Theorem 4.3 , the extension for Z 2 k 1 do es not split. By Theorem 3.3 , nonsplitting extends to an y group containing Z 2 k 1 as a direct summand; therefore [ O A 2 ]  = 0 . (B) A 2 is elementary ab elian of r ank ≥ 2 . Equiv alen tly , k i = 1 for all i and m ≥ 2 , and therefore A 2 ∼ = ( Z 2 ) m with m ≥ 2 . Then ( Z 2 ) 2 is a direct summand of A 2 . By Theorem 5.2 , the extension for ( Z 2 ) 2 do es not split, hence [ O ( Z 2 ) 2 ]  = 0 . Applying Theorem 3.3 yields [ O A 2 ]  = 0 for all m ≥ 2 . The t w o splitting cases are therefore A 2 = 1 and A 2 ∼ = Z 2 , while cases (A) and (B) cov er all groups with | A 2 | ≥ 4 and give nonsplitting. This prov es the splitting criterion. □ A cknowledgments The author was partially supp orted b y Gran t INV-2025-213-3452 from the School of Science of Univ ersidad de los Andes. References [1] S. Aaronson and D. Gottesman. Impro ved sim ulation of stabilizer circuits. Physic al R eview A , 70:052328, 2004. [2] D. M. Appleb y . SIC-POVMs and the extended Clifford group. Journal of Mathematical Physics , 46(5):052107, 2005. [3] B. Bolt, T. G. Ro om, and G. E. W all. On the Clifford collineation, transform and similarit y groups I. Journal of the A ustr alian Mathematic al So ciety , 2(1):60–79, 1961. [4] B. Bolt, T. G. Room, and G. E. W all. 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