# Second cohomology group for trivial group action of A5 on Z2

From Groupprops

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:

where is alternating group:A5 (the alternating group on a set of size five) and is cyclic group:Z2. Note that has order and 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 | -- | -- |