# Second cohomology group for nontrivial group action of Z2 on Z4

This article gives information about the second cohomology group for a specified nontrivial action of the group cyclic group:Z2 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 as an abelian normal subgroup and cyclic group:Z2 the corresponding quotient group.
The value of this cohomology group is cyclic group:Z2.
Get more specific information about cyclic group:Z2 |Get more specific information about cyclic group:Z4|View other constructions whose value is cyclic group:Z2

## Description of the group

Let $G$ be cyclic group:Z2 and $A$ be cyclic group:Z4. Note that $\operatorname{Aut} A$ is isomorphic to the group cyclic group:Z2, and it comprises the identity automorphism and the automorphism sending every element of $A$ to its inverse. Consider the homomorphism $\varphi:G \to \operatorname{Aut}(A)$ that is an isomorphism: it sends the non-identity element of $G$ to the inverse map on $A$, and the identity element of $G$ to the identity automorphism of $A$.

We are interested in the second cohomology group for the action $\varphi$ of $G$ on $A$, i.e., the group: $\! H^2_\varphi(G,A)$

The cohomology group is isomorphic to cyclic group:Z2.

## Elements

Let $g$ denote the non-identity element of $G$ and $a$ denote a generator for $A$. We consider here the two cohomology classes, with representative cocycles for each. For simplicity, we choose a representative cocycle that is a normalized 2-cocycle, i.e., if either of the inputs is the identity element of $G$, the output is the identity element of $A$. Thus, to specify the cocycle $f$, we need only specify $f(g,g)$.

Cohomology class type Number of cohomology classes Representative cocycle Corresponding group extension GAP ID (second part, order is 8) Base characteristic in whole group?
trivial 1 $f(g,g)$ takes the value zero, i.e., is the identity element of $A$. dihedral group:D8 3 Yes
nontrivial 1 $f(g,g) = a^2$ quaternion group 4 No

## Group actions

Because each of the cohomology class types has size one, thereis no scope for permutation of these under any group actions.