Second cohomology group for trivial group action of Z2 on Z4

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 cyclic group:Z2 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 in the center and cyclic group:Z2 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
Get more specific information about cyclic group:Z2 |Get more specific information about cyclic group:Z4

Description of the group

We consider here the second cohomology group for trivial group action of the cyclic group:Z2 on the cyclic group:Z4, i.e.,

$\!H^{2}(G,A)$

where $G\cong \mathbb {Z} _{2}$ and $A\cong \mathbb {Z} _{2}$.

The cohomology group is isomorphic to cyclic group:Z2.

Elements

We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

Cohomology class type	Number of cohomology classes	Representative 2-cocycle	Corresponding group extension	Second part of GAP ID (order is 8)
trivial	1	$0$ everywhere	direct product of Z4 and Z2	2
nontrivial	1	$f(1,1)=1$, all others $0$	cyclic group:Z8	1