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 <td style="vertical-align: top;"><a
href="aboutNonabelian.html">Previous</a> 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: A relative Schur multiplier,
Baer invariants 
and the capability of groups 
 </big></td>
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href="aboutArtinGroups.html">next</a> 
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">As
mentioned <a style="color: rgb(0, 0, 102);" href="aboutTopology.html">previously</a>,
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 
 
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Hn(G,N,Z) = Hn+1(B(G,N),Z) 
 
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for all n>0. The homology exact sequence of the cofibration can then
be written as 
 
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 ···
→ H3(Q,Z) → H2(G,N,Z) → H2(G,Z) → H2(Q,Z) → H1(G,N,Z) → H1(G,Z) → H1(Q,Z) → 0 . 
 
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 <div style="text-align: left;">There is an isomorphism 
 
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H1(G,N,Z) = N/[N,G] 
 
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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 
 
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style="text-align: center; vertical-align: top; background-color: rgb(204, 255, 255);"> 
 H2(G,N,Z) =
Ker( N ^ G → N ) , 
 
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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",
style="font-style: italic;">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
. </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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: 
 <ul>
 <li>ph( x )=f( x ) for all x in N,</li>
 <li>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 ).

 </li>
 </ul>
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!) </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">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.<big><big><big><big><big><big><big><big><big><big><big><big><big><big><big><big></big></big></big></big></big></big></big></big><big><big><big><big><big><big><big><big></big></big></big></big></big></big></big></big></big></big></big></big></big></big></big></big> 
 
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: 
 <ol>
 <li>d(gm) = gd( m )g-1
for g in G and m in M;</li>
 <li>m m' m^-1 = ^d(m)m
for m and m' in M;
 <li>N = Image( d) ;</li>
 <li>the action of G on M is such that G acts trivially on the
kernel of d.</li>
 </ol>
(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. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
G:=SylowSubgroup(MathieuGroup(24),2);; 
gap> N:=Centre(G);; 
gap> Order(EpiCentre(G,N)); 
1 
 
gap> Order(EpiCentre(G)); 
2 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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.) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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,<big><big>
On the nilpotent multipliers of a group, Mathematische Zeitschrifft 226
(1997), 405-428]. </big></big>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. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">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. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">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 ] 
 </td>
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