The Schur multiplier.

*(English)*Zbl 0619.20001
London Mathematical Society Monographs. New Series 2. Oxford: Clarendon Press. xiv, 302 p. £45.00 (1987).

The multipliers (now named Schur multiplier or multiplicator) were first introduced in the fundamental work of I. Schur [J. Reine Angew. Math. 127, 20–50 (1904; JFM 35.0155.01)] in order to study projective representations of groups. Since then Schur multipliers have proved to be an extremely powerful tool in many areas and a large amount of research has been devoted to the study of various properties of the Schur multipliers of a group \(G\). The Schur multiplier \(M(G)\) of a group \(G\) is also identified with the second cohomology group \(H^ 2(G,\mathbb{C}^*)\) of \(G\), where \(\mathbb{C}^*\) (respectively \(T\)) denotes the multiplicative group of the field of complex numbers (denotes the additive group of rationals mod 1) regarded as a trivial \(G\)-module. The author in the present book starts with the goal of giving a systematic (but in no way encyclopedic) account of the work done on the Schur multiplier during the last about thirty years the effort being to convey a comprehensive picture of the current state of the subject.

The first chapter sets up notations and terminology to be used in the rest of the book and also some standard results from the cohomology of groups are given for convenience of reference.

Chapter II begins with establishing some properties of the multiplicator of a finite group \(G\). Important among these are: If \(P\) is a Sylow \(p\)-subgroup of \(G\), then the \(G\)-stable subgroup \(M(P)^G\) of \(M(P)\) is a direct factor of \(M(P)\). The result of Schur that every finite group has at least one covering group is proved and then used to prove the result of (i) Schur again that \(e^2\) divides \(o(G)\) and (ii) Alperin and Tze-Non Kuo that \(ee'\) divides \(o(G)\), where \(e,e'\) denote the exponents of \(M(G)\) and \(G\) respectively. Another application of the covering group is the result of M. R. Jones and J. Wiegold [J. Lond. Math. Soc., II. Ser. 6, 738 (1973; Zbl 0258.20029)] that if \(H\) is a subgroup of index \(n\) in \(G\), then the group \(M(G)^n\) of all \(n\)th powers of \(M(G)\) is isomorphic to a subgroup of \(M(H)\). Let \(A\) be a finite Abelian group and \(A^*= \operatorname{Hom}(A,{\mathbb{C}}^*)\). Then there exists a splitting exact sequence \[ 0\to \mathrm{Ext}(G/G',A)\to H^ 2(G,A)\to \operatorname{Hom}(A^*,M(G))\to 0. \] An application of this result yields the result of K. Yamazaki [J. Fac. Sci., Univ. Tokyo, Sect. I 10, 147–195 (1964; Zbl 0125.01601)] that if \(1\to A\to H\to G\to 1\) is a stem extension of \(G\), then there exists a stem cover \(G^*\) of \(G\) such that \(H\) is a homomorphic image of \(G^*\).

Let \(G=N\rtimes T\) be the semi direct product of a normal subgroup \(N\) and a subgroup \(T\). Then \(M(T)\) is a direct factor of \(M(G)\). However, the complementary factor \(\tilde M(G)\) is not easy to compute explicitly but is given through an exact sequence of K. Tahara [Math. Z. 129, 365–379 (1972; Zbl 0238.20068)] \[ 1\to H^1(T,N^*)\to \widetilde M(G)\to M(N)^T\to H^2(T,N^*) \tag{1} \] where \(N^*= \operatorname{Hom}(N,\mathbb{C}^*)\). The complementary factor \(\widetilde M(G)\) is explicitly computed when \(G=N\times T\) and also as is clear from the exact sequence (1) \(\widetilde M(G)=M(N)^T\) when \((| N|,| T|)=1\).

There is a close link between \(M(G)\), projective representations of \(G\) over \(\mathbb{C}\) and twisted group algebras of \(G\) over \(\mathbb{C}\), where \(G\) is a finite group. In fact twisted group algebras \(\mathbb{C}^{\alpha}G\) and \(\mathbb{C}^{\beta}G\) are equivalent if and only if the 2-cocycles \(\alpha,\beta \in Z^2(G,\mathbb{C}^*)\) are cohomologous. Then the map \(\alpha \to \mathbb{C}^{\alpha}G\) induces a bijective correspondence between the elements of \(M(G)\) and the equivalence classes of twisted group algebras of \(G\) over \(\mathbb{C}\). Regarding representations, for any \(\alpha \in Z^2(G,\mathbb{C}^*)\), \(G\) has an irreducible \(\alpha\)-representation. This provides an important step in the proof of the exact sequence \(M(G/Z)\to M(G)\to G\otimes Z\) where \(Z\) denotes the centre of \(G\).

Let \(G=F/R\) be a free presentation of a finite group \(G\) with \(F\) free of rank \(n\). Then \(M(G)\cong R\cap F'/[F,R]\) is the torsion subgroup of \(R/[F,R]\). Moreover, if \(S\) is a normal subgroup of \(F\) such that \(R/[F,R]=R\cap F'/[F,R]\times S/[F,R]\), then \(S/[F,R]\) is free abelian of rank \(n\) and \(G^*=F/S\) is a covering group of \(G\). This then gives that if \(G\) is finitely generated by \(n\) elements with \(r\) defining relations and \(s\) is the minimum number of generators of \(G\), then \(r\geq n+s-a\) result extremely useful in computing \(M(G)\) for specific groups. These results are then used to get information on covering groups of \(G\), exponent of \(M(G)\) and a link between the structure of \(M(G/Z)\) and \(M(G)\) where \(Z\) is a central subgroup of the finite group \(G\). For example if \(G\) is a finite group, \(G^*\) a covering group of \(G\) then \(M(G^*)\) is isomorphic to a subgroup of \(G\otimes M(G)\).

For any group \(G\), let \(C(G)\) denote the group of commutator relations of \(G\) and \(B(G)\) the normal closure in \(<G,G>\) of the universal commutator relations. Then \[ H(G)=C(G)/B(G)\cong R\cap F'/[F,R]\cong H_2(G,Z) \] where \(G=F/R\) is a free presentation of \(G\). Moreover for arbitrary groups \(G_1\) and \(G_2\), \[ H(G_1*G_2)\cong H(G_1)\times H(G_2)\text{ while } H(G_1\times G_2)\cong H(G_1)\times H(G_2)\times G_1\otimes G_2. \] Suppose that \(G\) is a normal subgroup of a finite group \(G^*\) with \(G^*/G\) cyclic. Then there is an exact sequence \(M(G)\to M(G^*)\to H\to 0\) where \(H\) is a suitably defined quotient of a subgroup of \(G/G'\). This exact sequence is then used to obtain some upper bounds on the order and the minimum number of generators of \(M(G)\) for a finite group \(G\).

Let \(A,B\) be any finite groups \(A^*\), \(B^*\) any covering groups of \(A, B\) respectively. Then the second nilpotent product \(A^*\circ B^*\) is a covering group of \(A\times B\). A covering group of a finite abelian \(p\)-group is computed and if \(G\) is a perfect group with a free presentation \(F/R\), then \(F'/[F,R]\) is a covering group of \(G\). If \(A_n\) is the alternating group of degree \(n\) then \(M(A_n)\) and a covering group of \(A_n\) are computed explicitly.

The multiplicator \(M(G)\) of a finite group \(G\) is finite and a \(p\)-component of \(M(G)\), \(p\) a prime is isomorphic to a subgroup of \(M(P)\) where \(P\) is a Sylow \(p\)-subgroup of \(G\). Taking a cue from this chapter III is devoted to study the multiplicator of a finite \(p\)-group. There is a connection between the order of \(M(G)\) and order of the derived group \(G'\) of the finite \(p\)-group \(G\). If \(G\) is of order \(p^n\), nilpotence class \(c\) and it admits a presentation with \(d\) generators and \(r\) relations then \[ p^{d(d- 1)/2}\leq | M(G)| | G'| \leq p^{n(n-1)/2} \tag{i} \] and \[ d(M(G))\leq \sum^{c}_{j=1}m_d(j+1), \tag{ii} \] where \(m_d(q)\) denotes the rank of the free abelian group \(\gamma_q(F_d)/\gamma_{q+1}(F_d)\) with \(F_d\) a free group of rank \(d\). This bound for \(d(M(G))\) is best possible. The multiplicator \(M(G)\) when \(G\) is a \(p\)-group of class \(2\) and \(G/G'\) is elementary abelian admits an explicit description. In particular if \(G\) is an extra-special \(p\)-group of order \(p^{2n+1}\) then \(M(G)\) is an elementary Abelian \(p\)-group of order \(p^{2n^2-n-1}\) and if \(n>1\), \(G\) is unicentral. Some properties of \(p\)-groups \(G\) with trivial multiplicator are obtained. For example \(d(G)\geq 4\) for such a group \(G\).

Let \(G\) be a finite group, \(p\) a prime and \(P\) a fixed Sylow \(p\)-subgroup of \(G\). Then the normalizer \(N_G(Q)\) of a non-identity subgroup \(Q\) of \(P\) is called a \(p\)-local subgroup of \(G\) and by the \(p\)-local structure of \(G\) we understand the structure of all p-local subgroups of \(G\). The 4th chapter is devoted to examine conditions under which the \(p\)-structure of \(M(G)\) is determined by the \(p\)-local structure of \(G\). The main result here is the theorem of D. F. Holt [Q. J. Math., Oxf. II. Ser. 28, 495–508 (1977; Zbl 0373.20019)] that \(M(G)_p\cong M(N_G(P))_p\) provided nilpotency class of \(P\) is \(\leq p/2\).

Chapter V is devoted to examine connections between orthogonal \(\mathbb{C}G\)-modules, the cohomology group \(H^2(G,Z_2)\) and the Schur multiplicator \(M(G)\). These techniques are used to prove that (i) the Schur multiplier of the Hall-Janko group has even order; (ii) if \(G\) is a finite group containing a Sylow 2-subgroup \(S\) which is a direct sum of two cyclic groups, then \(M(G)\) has even order and (iii) if \(G\) is a group containing a dihedral Sylow 2-subgroup then the 2-component of \(M(G)\) has order 2. The main source for the results of this chapter being the fundamental paper [J. Algebra 74, 20–51 (1982; Zbl 0479.20005)] of S. Gagola and S. Garrison.

Let \(G\) be a group, \(\{A_i\}_{i\in I}\) be a family of subgroups where \(I\) is an ordered index set and for each \(i\in I\), let \(\hat A_i\) denote the product of normal closures of all the \(A_j\) in \(G\), \(j\in I\), \(j\neq i\). The group \(G\) is called a regular product of this family of subgroups if \(G\) is generated by the family of subgroups and \(A_i\cap \hat A_i=1\) for all \(i\in I\). Chapter VI studies the multiplicator of a regular product of groups. Explicitly, \[ M(G)\cong (\prod_{i\in I}^x M(A_i))\times H/[H,A] \] where \(A\) denotes the free product of the family of subgroups \(\{A_i\}_{i\in I}\) and \(H\) is the kernel of the natural homomorphism \(A\to G\) induced by the identity map on each \(A_i\), \(i\in I\). Following N. Blackburn [Ill. J. Math. 16, 116–129 (1972; Zbl 0242.20036)] the multiplicator of standard wreath products is computed: Let \(m\) denote the number of elements of order \(2\) in \(G\). Then the Schur multiplier \(M(G\wr H)\) of the standard wreath product \(G\wr H\) is the direct product of \(M(G)\), \(M(H)\), \((1/2)(| G|-m-1)\) copies of \(H\otimes H\) and \(m\) copies of \(H{\#}H\), where \(H{\#}H\) is the factor group of \(H\otimes H\) by the subgroup generated by elements of the form \(a\otimes b+b\otimes a\). The Sylow \(p\)-subgroup \(P_n\) of the symmetric group of degree \(p^n\) equals \(P_1\wr P_{n-1}\) for each \(n>1\) where \(P_1\) denotes a cyclic group of order \(p\). Using this representation of \(P_n\), \(M(P_n)\) is shown to be an elementary abelian \(p\)-group of order \(p^s\) where \(s\) is a specified number. Consequently any \(p\)-group can be embedded in a \(p\)-group whose multiplicator is elementary Abelian. Multipliers of the standard wreath products of some special groups are then explicitly computed.

Chapter VII is devoted to the computation of Schur multipliers of special and projective special linear groups, Coxeter groups, subgroup of rotations of Coxeter groups and Suzuki groups. The book concludes with a chapter giving tables of Schur multipliers of some individual groups.

In my opinion the book is a very welcome addition to the list of books already available on the subject.

The first chapter sets up notations and terminology to be used in the rest of the book and also some standard results from the cohomology of groups are given for convenience of reference.

Chapter II begins with establishing some properties of the multiplicator of a finite group \(G\). Important among these are: If \(P\) is a Sylow \(p\)-subgroup of \(G\), then the \(G\)-stable subgroup \(M(P)^G\) of \(M(P)\) is a direct factor of \(M(P)\). The result of Schur that every finite group has at least one covering group is proved and then used to prove the result of (i) Schur again that \(e^2\) divides \(o(G)\) and (ii) Alperin and Tze-Non Kuo that \(ee'\) divides \(o(G)\), where \(e,e'\) denote the exponents of \(M(G)\) and \(G\) respectively. Another application of the covering group is the result of M. R. Jones and J. Wiegold [J. Lond. Math. Soc., II. Ser. 6, 738 (1973; Zbl 0258.20029)] that if \(H\) is a subgroup of index \(n\) in \(G\), then the group \(M(G)^n\) of all \(n\)th powers of \(M(G)\) is isomorphic to a subgroup of \(M(H)\). Let \(A\) be a finite Abelian group and \(A^*= \operatorname{Hom}(A,{\mathbb{C}}^*)\). Then there exists a splitting exact sequence \[ 0\to \mathrm{Ext}(G/G',A)\to H^ 2(G,A)\to \operatorname{Hom}(A^*,M(G))\to 0. \] An application of this result yields the result of K. Yamazaki [J. Fac. Sci., Univ. Tokyo, Sect. I 10, 147–195 (1964; Zbl 0125.01601)] that if \(1\to A\to H\to G\to 1\) is a stem extension of \(G\), then there exists a stem cover \(G^*\) of \(G\) such that \(H\) is a homomorphic image of \(G^*\).

Let \(G=N\rtimes T\) be the semi direct product of a normal subgroup \(N\) and a subgroup \(T\). Then \(M(T)\) is a direct factor of \(M(G)\). However, the complementary factor \(\tilde M(G)\) is not easy to compute explicitly but is given through an exact sequence of K. Tahara [Math. Z. 129, 365–379 (1972; Zbl 0238.20068)] \[ 1\to H^1(T,N^*)\to \widetilde M(G)\to M(N)^T\to H^2(T,N^*) \tag{1} \] where \(N^*= \operatorname{Hom}(N,\mathbb{C}^*)\). The complementary factor \(\widetilde M(G)\) is explicitly computed when \(G=N\times T\) and also as is clear from the exact sequence (1) \(\widetilde M(G)=M(N)^T\) when \((| N|,| T|)=1\).

There is a close link between \(M(G)\), projective representations of \(G\) over \(\mathbb{C}\) and twisted group algebras of \(G\) over \(\mathbb{C}\), where \(G\) is a finite group. In fact twisted group algebras \(\mathbb{C}^{\alpha}G\) and \(\mathbb{C}^{\beta}G\) are equivalent if and only if the 2-cocycles \(\alpha,\beta \in Z^2(G,\mathbb{C}^*)\) are cohomologous. Then the map \(\alpha \to \mathbb{C}^{\alpha}G\) induces a bijective correspondence between the elements of \(M(G)\) and the equivalence classes of twisted group algebras of \(G\) over \(\mathbb{C}\). Regarding representations, for any \(\alpha \in Z^2(G,\mathbb{C}^*)\), \(G\) has an irreducible \(\alpha\)-representation. This provides an important step in the proof of the exact sequence \(M(G/Z)\to M(G)\to G\otimes Z\) where \(Z\) denotes the centre of \(G\).

Let \(G=F/R\) be a free presentation of a finite group \(G\) with \(F\) free of rank \(n\). Then \(M(G)\cong R\cap F'/[F,R]\) is the torsion subgroup of \(R/[F,R]\). Moreover, if \(S\) is a normal subgroup of \(F\) such that \(R/[F,R]=R\cap F'/[F,R]\times S/[F,R]\), then \(S/[F,R]\) is free abelian of rank \(n\) and \(G^*=F/S\) is a covering group of \(G\). This then gives that if \(G\) is finitely generated by \(n\) elements with \(r\) defining relations and \(s\) is the minimum number of generators of \(G\), then \(r\geq n+s-a\) result extremely useful in computing \(M(G)\) for specific groups. These results are then used to get information on covering groups of \(G\), exponent of \(M(G)\) and a link between the structure of \(M(G/Z)\) and \(M(G)\) where \(Z\) is a central subgroup of the finite group \(G\). For example if \(G\) is a finite group, \(G^*\) a covering group of \(G\) then \(M(G^*)\) is isomorphic to a subgroup of \(G\otimes M(G)\).

For any group \(G\), let \(C(G)\) denote the group of commutator relations of \(G\) and \(B(G)\) the normal closure in \(<G,G>\) of the universal commutator relations. Then \[ H(G)=C(G)/B(G)\cong R\cap F'/[F,R]\cong H_2(G,Z) \] where \(G=F/R\) is a free presentation of \(G\). Moreover for arbitrary groups \(G_1\) and \(G_2\), \[ H(G_1*G_2)\cong H(G_1)\times H(G_2)\text{ while } H(G_1\times G_2)\cong H(G_1)\times H(G_2)\times G_1\otimes G_2. \] Suppose that \(G\) is a normal subgroup of a finite group \(G^*\) with \(G^*/G\) cyclic. Then there is an exact sequence \(M(G)\to M(G^*)\to H\to 0\) where \(H\) is a suitably defined quotient of a subgroup of \(G/G'\). This exact sequence is then used to obtain some upper bounds on the order and the minimum number of generators of \(M(G)\) for a finite group \(G\).

Let \(A,B\) be any finite groups \(A^*\), \(B^*\) any covering groups of \(A, B\) respectively. Then the second nilpotent product \(A^*\circ B^*\) is a covering group of \(A\times B\). A covering group of a finite abelian \(p\)-group is computed and if \(G\) is a perfect group with a free presentation \(F/R\), then \(F'/[F,R]\) is a covering group of \(G\). If \(A_n\) is the alternating group of degree \(n\) then \(M(A_n)\) and a covering group of \(A_n\) are computed explicitly.

The multiplicator \(M(G)\) of a finite group \(G\) is finite and a \(p\)-component of \(M(G)\), \(p\) a prime is isomorphic to a subgroup of \(M(P)\) where \(P\) is a Sylow \(p\)-subgroup of \(G\). Taking a cue from this chapter III is devoted to study the multiplicator of a finite \(p\)-group. There is a connection between the order of \(M(G)\) and order of the derived group \(G'\) of the finite \(p\)-group \(G\). If \(G\) is of order \(p^n\), nilpotence class \(c\) and it admits a presentation with \(d\) generators and \(r\) relations then \[ p^{d(d- 1)/2}\leq | M(G)| | G'| \leq p^{n(n-1)/2} \tag{i} \] and \[ d(M(G))\leq \sum^{c}_{j=1}m_d(j+1), \tag{ii} \] where \(m_d(q)\) denotes the rank of the free abelian group \(\gamma_q(F_d)/\gamma_{q+1}(F_d)\) with \(F_d\) a free group of rank \(d\). This bound for \(d(M(G))\) is best possible. The multiplicator \(M(G)\) when \(G\) is a \(p\)-group of class \(2\) and \(G/G'\) is elementary abelian admits an explicit description. In particular if \(G\) is an extra-special \(p\)-group of order \(p^{2n+1}\) then \(M(G)\) is an elementary Abelian \(p\)-group of order \(p^{2n^2-n-1}\) and if \(n>1\), \(G\) is unicentral. Some properties of \(p\)-groups \(G\) with trivial multiplicator are obtained. For example \(d(G)\geq 4\) for such a group \(G\).

Let \(G\) be a finite group, \(p\) a prime and \(P\) a fixed Sylow \(p\)-subgroup of \(G\). Then the normalizer \(N_G(Q)\) of a non-identity subgroup \(Q\) of \(P\) is called a \(p\)-local subgroup of \(G\) and by the \(p\)-local structure of \(G\) we understand the structure of all p-local subgroups of \(G\). The 4th chapter is devoted to examine conditions under which the \(p\)-structure of \(M(G)\) is determined by the \(p\)-local structure of \(G\). The main result here is the theorem of D. F. Holt [Q. J. Math., Oxf. II. Ser. 28, 495–508 (1977; Zbl 0373.20019)] that \(M(G)_p\cong M(N_G(P))_p\) provided nilpotency class of \(P\) is \(\leq p/2\).

Chapter V is devoted to examine connections between orthogonal \(\mathbb{C}G\)-modules, the cohomology group \(H^2(G,Z_2)\) and the Schur multiplicator \(M(G)\). These techniques are used to prove that (i) the Schur multiplier of the Hall-Janko group has even order; (ii) if \(G\) is a finite group containing a Sylow 2-subgroup \(S\) which is a direct sum of two cyclic groups, then \(M(G)\) has even order and (iii) if \(G\) is a group containing a dihedral Sylow 2-subgroup then the 2-component of \(M(G)\) has order 2. The main source for the results of this chapter being the fundamental paper [J. Algebra 74, 20–51 (1982; Zbl 0479.20005)] of S. Gagola and S. Garrison.

Let \(G\) be a group, \(\{A_i\}_{i\in I}\) be a family of subgroups where \(I\) is an ordered index set and for each \(i\in I\), let \(\hat A_i\) denote the product of normal closures of all the \(A_j\) in \(G\), \(j\in I\), \(j\neq i\). The group \(G\) is called a regular product of this family of subgroups if \(G\) is generated by the family of subgroups and \(A_i\cap \hat A_i=1\) for all \(i\in I\). Chapter VI studies the multiplicator of a regular product of groups. Explicitly, \[ M(G)\cong (\prod_{i\in I}^x M(A_i))\times H/[H,A] \] where \(A\) denotes the free product of the family of subgroups \(\{A_i\}_{i\in I}\) and \(H\) is the kernel of the natural homomorphism \(A\to G\) induced by the identity map on each \(A_i\), \(i\in I\). Following N. Blackburn [Ill. J. Math. 16, 116–129 (1972; Zbl 0242.20036)] the multiplicator of standard wreath products is computed: Let \(m\) denote the number of elements of order \(2\) in \(G\). Then the Schur multiplier \(M(G\wr H)\) of the standard wreath product \(G\wr H\) is the direct product of \(M(G)\), \(M(H)\), \((1/2)(| G|-m-1)\) copies of \(H\otimes H\) and \(m\) copies of \(H{\#}H\), where \(H{\#}H\) is the factor group of \(H\otimes H\) by the subgroup generated by elements of the form \(a\otimes b+b\otimes a\). The Sylow \(p\)-subgroup \(P_n\) of the symmetric group of degree \(p^n\) equals \(P_1\wr P_{n-1}\) for each \(n>1\) where \(P_1\) denotes a cyclic group of order \(p\). Using this representation of \(P_n\), \(M(P_n)\) is shown to be an elementary abelian \(p\)-group of order \(p^s\) where \(s\) is a specified number. Consequently any \(p\)-group can be embedded in a \(p\)-group whose multiplicator is elementary Abelian. Multipliers of the standard wreath products of some special groups are then explicitly computed.

Chapter VII is devoted to the computation of Schur multipliers of special and projective special linear groups, Coxeter groups, subgroup of rotations of Coxeter groups and Suzuki groups. The book concludes with a chapter giving tables of Schur multipliers of some individual groups.

In my opinion the book is a very welcome addition to the list of books already available on the subject.

Reviewer: L. R. Vermani (Kurukshetra)

##### MSC:

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20J05 | Homological methods in group theory |

20C25 | Projective representations and multipliers |

20D15 | Finite nilpotent groups, \(p\)-groups |

20J06 | Cohomology of groups |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |