Second cohomology group for trivial group action of A5 on Z2

From Groupprops
Jump to: navigation, search
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:A5 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and alternating group:A5 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is cyclic group:Z2.
Get more specific information about alternating group:A5 |Get more specific information about cyclic group:Z2|View other constructions whose value is cyclic group:Z2

Description of the group

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

\! H^2(G;A)

where G \cong A_5 is alternating group:A5 (the alternating group on a set of size five) and A is cyclic group:Z2. Note that G has order 5!/2 = 60 and A has order 2.

The cohomology group itself is isomorphic to cyclic group:Z2.

Computation of the group

The group can be computed using group cohomology of alternating group:A5#Cohomology groups for trivial group action.

Elements

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 120)
trivial 1 direct product of A5 and Z2 35
nontrivial 1 special linear group:SL(2,5) 5
Total (--) 2 -- --