Group cohomology of alternating group:A4

This article gives specific information, namely, group cohomology, about a particular group, namely: alternating group:A4.
View group cohomology of particular groups | View other specific information about alternating group:A4

Family contexts

Family name	Parameter value	Information on group cohomology of family
alternating group $A_n$ of degree $n$	$n = 4$, i.e., the group $A_4$	group cohomology of alternating groups
projective special linear group of degree two over a finite field of size $q$, denoted $PSL(2,q)$	[[$q = 3$, i.e., field:F3, so the group is $PSL(2,3)$.	group cohomology of projective special linear group of degree two over a finite field

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

The general description is unclear. The group is not a finite group with periodic cohomology, but there is probably a description that involves a variable power of an abelian group direct summed with another abelian group.

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The first few homology groups are given below:

$q$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$H_q$	$\mathbb{Z}$	$\mathbb{Z}/3\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/6\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/6\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/6\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$

GAP implementation

Computation of integral homology

The homology groups for trivial group action with coefficients in $\mathbb{Z}$ can be computed in GAP using the GroupHomology function in the HAP package, which can be loaded by the command LoadPackage("hap"); if it is installed but not loaded. The function outputs the orders of cyclic groups for which the homology or cohomology group is the direct product of these (more technically, it outputs the elementary divisors for the homology or cohomology group that we are trying to compute).

Here are computations of the first few homology groups:

Computation of first homology group

gap> GroupHomology(AlternatingGroup(4),1);
[ 3 ]

This says that the abelianization (which is also the first homology group for trivial group action on the integers) is cyclic group:Z3, which we already know from the fact that the derived subgroup is V4 in A4.

Computation of second homology group

gap> GroupHomology(AlternatingGroup(4),2);
[ 2 ]

This says that the Schur multiplier (the second homology group for trivial group action on the integers) is cyclic group:Z2.

Computation of first few homology groups=

gap> List([1,2,3,4,5,6,7,8],i->[i,GroupHomology(AlternatingGroup(4),i)]);
[ [ 1, [ 3 ] ], [ 2, [ 2 ] ], [ 3, [ 6 ] ], [ 4, [  ] ], [ 5, [ 2, 6 ] ], 
  [ 6, [ 2 ] ], [ 7, [ 6 ] ], [ 8, [ 2, 2 ] ] ]

Here's a more comprehensive listing:

gap> List([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],i->[i,GroupHomology(AlternatingGroup(4),i)]);
[ [ 1, [ 3 ] ], [ 2, [ 2 ] ], [ 3, [ 6 ] ], [ 4, [  ] ], [ 5, [ 2, 6 ] ], 
  [ 6, [ 2 ] ], [ 7, [ 6 ] ], [ 8, [ 2, 2 ] ], [ 9, [ 2, 6 ] ], 
  [ 10, [ 2 ] ], [ 11, [ 2, 2, 6 ] ], [ 12, [ 2, 2 ] ], [ 13, [ 2, 6 ] ], 
  [ 14, [ 2, 2, 2 ] ], [ 15, [ 2, 2, 6 ] ], [ 16, [ 2, 2 ] ], 
  [ 17, [ 2, 2, 2, 6 ] ], [ 18, [ 2, 2, 2 ] ], [ 19, [ 2, 2, 6 ] ], 
  [ 20, [ 2, 2, 2, 2 ] ], [ 21, [ 2, 2, 2, 6 ] ], [ 22, [ 2, 2, 2 ] ], 
  [ 23, [ 2, 2, 2, 2, 6 ] ], [ 24, [ 2, 2, 2, 2 ] ] ]