


As
mentioned previously,
we can define H_{n}(G,Z) = H_{n}(B(G),Z) where B(G) is
any CWspace with fundamental group equal to G and for which all
other homotopy groups are trivial. Given a short exact sequence of
groups 1 → N → G → Q → 1 we set B(G,N) equal to the cofibre of the
induced cofibration B(G) → B(Q), and we define
for all n>0. The homology exact sequence of the cofibration can then be written as
There is an isomorphism
and textbooks often refer to the first five terms of the cofibration exact sequence, with third term replaced by N/[N,G], as the fiveterm HochschildSerre exact sequence (since these five terms can also be derived from the HochschildSerre spectral sequence for group extensions). Less wellknown is that, in light of an isomorphism
the first eight terms of the cofibration sequence are in fact a useful computational tool. The isomorphism for H_{2}(G,N,Z) involves a nonabelian exterior product (a quotient of the nonabelian tensor product of the previous page) and was proved by a topological argument in [R. Brown & J.L. Loday, "van Kampen theorems diagrams for diagrams of spaces", Topology 1987] and by an algebraic argument in [G. Ellis, "Nonabelian exterior products of groups and an exact sequence in the homology of groups", Glasgow Math. J. 29 (1987), 1319]. For a finite group G we refer to the homology group H_{2}(G,N,Z) as the relative Schur multiplier. When N=G this is the usual Schur multiplier H_{2}(G,G,Z) = H_{2}(G,Z). The following commands show that, for G the Sylow 2subgroup of the Mathieu group M_{24} and N its commutator subgroup, the relative Schur multiplier is H_{2}(G,N,Z) = (Z_{2})^{12} . 

gap>
G:=SylowSubgroup(MathieuGroup(24),2);; gap> N:=DerivedSubgroup(G);; gap> RelativeSchurMultiplier(G,N); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 

For
a finite group G the Universal Coefficient Theorem implies an
isomorphism H_{2}(G,Z) = H^{2}(G,C^{×}) where C^{×} is the group of
nonzero complex numbers. The group H^{2}(G,C^{×}) first appeared
in work of Schur on complex projective representations G → PGL(C). He proved, for example, that
every projective representation of G lifts to a linear representation G
→ GL(C) if and only if H^{2}(G,C^{×}) = 0. The relative Schur multiplier has a similar interpretation (though it does not seem to be recorded anywhere in the literature). Let p:GL(C) → PGL(C) be the canonical projection and note that Ker(p) is central in GL(C). Suppose that N is normal in G and that we have a homomorphism f:G → PGL(C). Let us say that a homomorphism h:N → GL(C) is a relative lift of f if:


A
second application of the (relative) Schur multiplier concerns groups G
that are isomorphic to a quotient G = K/Z(K) of a group K by the centre
of K. Such groups G are said to be capable.
The notion first arose in Philip Halls' work on classification of
pgroups.
Subsequently Beyl, Felgner and Schmid showed that, using the Schur
multiplier, one
can define a characteristic subgroup Z^{*}(G) of the centre of
G with the property that Z^{*}(G)=0 if and only if G is
capable. For details, see the paper [F.R. Beyl, U. Felgner and P.
Schmid, "On groups occuring as central factor groups", J. Algebra 61 (1979), 161177] .
The group Z^{*}(G) has recently become known as the epicentre of G. More generally, given a normal subgroup N in G, a relative central extension of the pair (G,N) consists of a group homomorphism d:M → G and action (g,m) → ^{g}m of G on M satisfying:
The following commands show that, for G the sylow 2subgroup of the Mathieu group M_{24} and N equal to the centre of G, the pair (G,N) is capable. They also show that the group G itself is not capable. 

gap>
G:=SylowSubgroup(MathieuGroup(24),2);; gap> N:=Centre(G);; gap> Order(EpiCentre(G,N)); 1 gap> Order(EpiCentre(G)); 2 

The
following commands quantify the number of capable primepower groups of
order
less than 256. (No cyclic group is capable, so prime orders are
omitted.) 

gap>
for i in [1..255] do > if IsPrimePowerInt(i) and not IsPrimeInt(i) then > NumberCapableGroups:=0; > for G in AllSmallGroups(i) do > if Order(EpiCentre(G))=1 then NumberCapableGroups:=NumberCapableGroups+1; > fi; > od; > Print("There are ",NumberSmallGroups(i), " groups of order ", i, " of which ", > NumberCapableGroups, " are capable. \n"); > fi; > od; There are 2 groups of order 4 of which 1 are capable. There are 5 groups of order 8 of which 2 are capable. There are 2 groups of order 9 of which 1 are capable. There are 14 groups of order 16 of which 5 are capable. There are 2 groups of order 25 of which 1 are capable. There are 5 groups of order 27 of which 2 are capable. There are 51 groups of order 32 of which 15 are capable. There are 2 groups of order 49 of which 1 are capable. There are 267 groups of order 64 of which 69 are capable. There are 15 groups of order 81 of which 5 are capable. There are 2 groups of order 121 of which 1 are capable. There are 5 groups of order 125 of which 2 are capable. There are 2328 groups of order 128 of which 432 are capable. There are 2 groups of order 169 of which 1 are capable. There are 67 groups of order 243 of which 19 are capable. gap> time; 246268 

A
number of papers have been written recently on the characterization of
capable pgroups. For example, the capable 2generator pgroups of
class two are classified for odd primes in [M. Bacon & L.C. Kappe,
On capable pgroups of nilpotency class two, Illinois J. Math 47 (2003) no.1/2,
4962] and for p=2 in [A. Magidin, Capable twogenerator 2groups, Communications in Algebra, to
appear.] These two papers also provide a good introduction to the
subject. The term epicentre, along with that of upper epicentral series, was coined in the paper [J.Burns & G.Ellis, On the nilpotent multipliers of a group, Mathematische Zeitschrifft 226 (1997), 405428]. The upper epicentral series 1 < Z_{1}^{*}(G) < Z_{2}^{*}(G) < ... is defined by setting Z_{c}^{*}(G) equal to the image in G of the cth term Z_{c}(U) of the upper central series of the group U=F/[[[R,F],F]...] (with c copies of F in the denominator) where F/R is any free presentation of G. It is not difficult to show that Z_{c}^{*}(G) is an invariant of G. We define a group G to be ccapable if it is isomorphic to a group K/Z(K) where the group K is (c1)capable; it is 1capable if it is capable. Note that if G is ccapable then it is also (c1)capable. It can be shown that a group G is ccapable if and only if Z_{c}^{*}(G)=1. It can also be shown that a finitely generated abelian group is ccapable (for any c) if and only if it is capable. The following commands quantify the number of 2capable primepower groups of order less than 256. Note that some groups are capable but not 2capable. 

gap>
for i in [1..255] do > if IsPrimePowerInt(i) and not IsPrimeInt(i) then > NumberCapableGroups:=0; > for G in AllSmallGroups(i) do > if Order(UpperEpicentralSeries(G,2))=1 then > NumberCapableGroups:=NumberCapableGroups+1; > fi; > od; > Print("There are ",NumberSmallGroups(i), " groups of order ", i, " of which ", > NumberCapableGroups, " are 2capable. \n"); > fi; > od; There are 2 groups of order 4 of which 1 are 2capable. There are 5 groups of order 8 of which 2 are 2capable. There are 2 groups of order 9 of which 1 are 2capable. There are 14 groups of order 16 of which 5 are 2capable. There are 2 groups of order 25 of which 1 are 2capable. There are 5 groups of order 27 of which 2 are 2capable. There are 51 groups of order 32 of which 14 are 2capable. There are 2 groups of order 49 of which 1 are 2capable. There are 267 groups of order 64 of which 58 are 2capable. There are 15 groups of order 81 of which 5 are 2capable. There are 2 groups of order 121 of which 1 are 2capable. There are 5 groups of order 125 of which 2 are 2capable. There are 2328 groups of order 128 of which 264 are 2capable. There are 2 groups of order 169 of which 1 are 2capable. There are 67 groups of order 243 of which 15 are 2capable. 

There
is a "dual" invariant to Z_{c}^{*}(G) which we denote
by M^{(c)}(G) and refer to as a Baer invariant. It is an abelian
group defined as the kernel of the canonical homomorphism L_{c+1}(U)
→ G where L_{c+1}(U) is the (c+1)st term of the lower central
series of the group U=F/[[[R,F],F]...] (with c
copies of F in the denominator)
where F/R is any free presentation of G. The invariant M^{(1)}(G)
is isomorphic to the second integral homology H_{2}(G,Z). Baer invariants can be defined for arbitrary varieties of groups, and so strictly speaking we should refer to M^{(c)}(G) as the Baer invariant of G with respect to the variety of nilpotent groups of class c. Note however that in the definition the group G itself need not be nilpotent. The following commands show that the Baer invariants M^{(c)}(H) for the Heisenberg group H on three complex variables, with C=1,2,3, are free abelian of ranks 14, 70 and 315 respectively. 

gap>
BaerInvariant(HeisenbergPcpGroup(3),1); [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> BaerInvariant(HeisenbergPcpGroup(3),2); [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> BaerInvariant(HeisenbergPcpGroup(3),3); [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 

