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 About HAP: A relative Schur multiplier, Baer invariants
and the capability of groups
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As mentioned previously, we can define Hn(G,Z) = Hn(B(G),Z) where B(G) is any CW-space 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


Hn(G,N,Z) = Hn+1(B(G,N),Z)

for all n>0. The homology exact sequence of the cofibration can then be written as


··· → H3(Q,Z) → H2(G,N,Z) → H2(G,Z) → H2(Q,Z) → H1(G,N,Z) → H1(G,Z) → H1(Q,Z) → 0 .

There is an isomorphism


H1(G,N,Z) = N/[N,G]

and textbooks often refer to the first five terms of the cofibration exact sequence, with third term replaced by N/[N,G], as the five-term Hochschild-Serre exact sequence (since these five terms can also be derived from the Hochschild-Serre spectral sequence for group extensions). Less well-known is that, in light of an isomorphism


H2(G,N,Z) = Ker( N ^ G → N ) ,

the first eight terms of the cofibration sequence are in fact a useful computational tool. The isomorphism for H2(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), 13-19].

For a finite group G we refer to the homology group H2(G,N,Z) as the relative Schur multiplier. When N=G this is the usual Schur multiplier H2(G,G,Z) = H2(G,Z). The following commands show that, for G the Sylow 2-subgroup of the Mathieu group M24 and N its commutator subgroup, the relative Schur multiplier is H2(G,N,Z) = (Z2)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 H2(G,Z) = H2(G,C×) where C× is the group of non-zero complex numbers. The group H2(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 H2(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:
  • ph( x )=f( x ) for all x in N,
  • h( gxg-1 )=   g' h( x ) g'-1 for all x in N and all g' in GL(C), g in G satisfying p( g' ) = f ( g ). 
Every projective representation f:G → PGL(C) admits a relative lift h:N → GL(C) if and only if H2(G,N,Z) = 0. (At least, I think this is the correct statement!)
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 p-groups. 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), 161-177] . 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) → gm of G on M satisfying:
  1. d(gm) = gd( m )g-1     for g in G and m in M;
  2. m m' m-1 = d(m) m      for m and m' in M;
  3. N = Image( d) ;
  4. the action of G on M is such that G acts trivially on the kernel of d.
(Conditions 1 and 2 assert that d:M → G is a crossed module.) The pair (G,N) is said to be capable if it admits a relative central extension with the property that Ker( d ) consists precisely of those elements in M on which G acts trivially. Using the relative Schur multiplier one can define a subgroup Z*(G,N) of the centre of N with the property that Z*(G,N) = 0 if and only if the pair (G,N) is capable. When N=G the group Z*(G,G) coincides with the epicenter Z*(G) of Beyl, Felgner and Schmid.

The following commands show that, for G the sylow 2-subgroup of the Mathieu group M24 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 prime-power 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 p-groups. For example, the capable 2-generator p-groups of class two are classified for odd primes in [M. Bacon & L.-C. Kappe, On capable p-groups of nilpotency class two, Illinois J. Math 47 (2003) no.1/2, 49-62] and for p=2 in [A. Magidin, Capable two-generator 2-groups, 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), 405-428]. The upper epicentral series 1 < Z1*(G) < Z2*(G) < ...  is defined by setting Zc*(G) equal to the image in G of the c-th term Zc(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 Zc*(G) is an invariant of G.

We define a group G to be c-capable if it is isomorphic to a group K/Z(K) where the group K is (c-1)-capable; it is 1-capable if it is capable. Note that if G is c-capable then it is also (c-1)-capable. It can be shown that a group G is c-capable if and only if  Zc*(G)=1. It can also be shown that a finitely generated abelian group is c-capable (for any c) if and only if it is capable.

The following commands quantify the number of 2-capable prime-power groups of order less than 256. Note that some groups are capable but not 2-capable. 
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 2-capable. \n");
> fi;
> od;
There are 2 groups of order 4 of which 1  are 2-capable.
There are 5 groups of order 8 of which 2  are 2-capable.
There are 2 groups of order 9 of which 1  are 2-capable.
There are 14 groups of order 16 of which 5  are 2-capable.
There are 2 groups of order 25 of which 1  are 2-capable.
There are 5 groups of order 27 of which 2  are 2-capable.
There are 51 groups of order 32 of which 14  are 2-capable.
There are 2 groups of order 49 of which 1  are 2-capable.
There are 267 groups of order 64 of which 58  are 2-capable.
There are 15 groups of order 81 of which 5  are 2-capable.
There are 2 groups of order 121 of which 1  are 2-capable.
There are 5 groups of order 125 of which 2  are 2-capable.
There are 2328 groups of order 128 of which 264  are 2-capable.
There are 2 groups of order 169 of which 1  are 2-capable.
There are 67 groups of order 243 of which 15  are 2-capable.
There is a "dual" invariant to Zc*(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 Lc+1(U) → G where Lc+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 H2(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 ]
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