# Second cohomology group for trivial group action of V4 on V4

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

## Description of the group

We consider here the second cohomology group for trivial group action of the Klein four-group on the Klein four-group, i.e., $\! H^2(G,A)$

where $G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ and $A \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.

The cohomology group is isomorphic to elementary abelian group:E64.

## Elements

FACTS TO CHECK AGAINST (second cohomology group for trivial group action):
Background reading on relationship with extension groups: Group extension problem
Arithmetic functions of extension group:
order (thus all extension groups have the same order): order of extension group is product of order of normal subgroup and quotient group
nilpotency class: nilpotency class of extension group is between nilpotency class of quotient group and one more for central extension
derived length: derived length of extension group is bounded by sum of derived length of normal subgroup and quotient group
minimum size of generating set: minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group|minimum size of generating set of quotient group is at most minimum size of generating set of group
WHAT'S THE TABLE BELOW?: Recall that there is a correspondence:
Elements of the group $H^2(G;A)$ for the trivial group action $\leftrightarrow$ congruence classes of central extensions with the specified subgroup $A$ and quotient group $G$.
This descends to a correspondence:
Orbits for the group action of $\operatorname{Aut}(G) \times \operatorname{Aut}(A)$ on $H^2(G;A)$ $\leftrightarrow$ pseudo-congruence classes of central extensions.
The table below breaks down the second cohomology group as a union of these orbits, with (as a general rule) each row describing one orbit, i.e., one "cohomology class type", aka one "pseudo-congruence class" of central extensions. The number of rows is the number of pseudo-congruence classes of central extensions.

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is 16) Stem extension? Base characteristic in whole group? Hall-Senior family (equivalence class up to being isoclinic) Nilpotency class of whole group (at least 2, at most 3) Derived length of whole group (always exactly 2) Minimum size of generating set of whole group (at least 2, at most 4) Subgroup information on base in whole group
trivial 1 elementary abelian group:E16 14 No No $\Gamma_1$ 1 1 4
nontrivial and symmetric 6 direct product of Z4 and Z4 2 No Yes $\Gamma_1$ 1 1 2
nontrivial and symmetric 9 direct product of Z4 and V4 10 No No $\Gamma_1$ 1 1 3
non-symmetric 18 SmallGroup(16,3) 3 No Yes $\Gamma_2$ 2 2 2 center of SmallGroup(16,3)
non-symmetric 18 nontrivial semidirect product of Z4 and Z4 4 No Yes $\Gamma_2$ 2 2 2 center of nontrivial semidirect product of Z4 and Z4
non-symmetric 9 direct product of D8 and Z2 11 No Yes $\Gamma_2$ 2 2 3 center of direct product of D8 and Z2
non-symmetric 3 direct product of Q8 and Z2 12 No Yes $\Gamma_2$ 2 2 3 center of direct product of Q8 and Z2
Total (7 rows) 64 -- -- -- -- $\Gamma_1, \Gamma_2$ -- -- -- --

## Subgroups of interest

Subgroup Value Corresponding group extensions for subgroup GAP IDs second part Group extension groupings for each coset GAP IDs second part
IIP subgroup of second cohomology group for trivial group action trivial group elementary abelian group:E16 (1 copy) 14 no groupings, cosets of size one 14, 2, 10, 3, 4, 11, 12
cyclicity-preserving subgroup of second cohomology group for trivial group action trivial group elementary abelian group:E16 (1 copy) 14 no groupings, cosets of size one 14, 2, 10, 3, 4, 11, 12
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions) elementary abelian group:E16 elementary abelian group:E16 (1 copy), direct product of Z4 and Z4 (3 copies) direct product of Z4 and V4 (9 copies) 14, 2, 10  ?  ?