Second cohomology group for trivial group action of A4 on Z2

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This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group alternating group:A4 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and alternating group:A4 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
Get more specific information about alternating group:A4 |Get more specific information about cyclic group:Z2

Description of the group

This article describes the second cohomology group for trivial group action:

\! H^2(G;A)

where G is alternating group:A4 (i.e., the alternating group on a set of size four) and A is cyclic group:Z2.

The cohomology group is isomorphic to cyclic group:Z2.

Computation in terms of group cohomology

By the formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization, we have that:

H^2(G;A) \cong \operatorname{Ext}^1(G^{\operatorname{ab}},A) \oplus \operatorname{Hom}(H_2(G;\mathbb{Z}),A)

For G the alternating group of degree four, we have, by group cohomology of alternating group:A4, that G^{\operatorname{ab}} (the abelianization, also the first homology group) is cyclic group:Z3 and H_2(G;\mathbb{Z}) (the Schur multiplier) is cyclic group:Z2. Plugging in, we get:

H^2(G;A) \cong \operatorname{Ext}^1(\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/2\mathbb{Z}) \oplus \operatorname{Hom}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}

Elements

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 24)
trivial 1 direct product of A4 and Z2 13
nontrivial 1 special linear group:SL(2,3) 3

Cocycles and coboundaries

Size information

We first give some quantitative size information if we use non-normalized cocycles and coboundaries:

Group Dimension as vector space over field:F2 Order of group (equals 2 to the power of dimension) Isomorphism class of group Explanation
group of 1-cocycles for trivial group action Z^1(G;A) 0 1 Klein four-group Same as \operatorname{Hom}(G,A). see first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
group of all 1-cochains for trivial group action C^1(G;A) 12 4096 elementary abelian group of order 4096 all set maps from G to A with pointwise addition, so the dimension is the cardinality of G.
group of all 2-coboundaries for trivial group action B^2(G;A) 12 4096 elementary abelian group of order 4096 By the first isomorphism theorem and the definition of this group, it is isomorphic to the group (1-cochains)/(1-cocycles), so the dimensions as vector spaces subtract and the orders divide.
group of all 2-cocycles for trivial group action Z^2(G;A) 13 8192 elementary abelian group of order 8192
second cohomology group for trivial group action 1 2 cyclic group:Z2 This is Z^2/B^2, so dimensions subtract and orders divide.