Second cohomology group for trivial Lie ring action of V4 on Z2

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This article gives information about the second cohomology group for trivial Lie ring action (i.e., the second cohomology group for Lie rings with trivial action) of the Lie ring Klein four-group on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and Klein four-group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base Lie ring and acting Lie ring.
The value of this cohomology group is elementary abelian group:E8.
Get more specific information about Klein four-group |Get more specific information about cyclic group:Z2|View other constructions whose value is elementary abelian group:E8

Description of the group

We consider here the second cohomology group for trivial Lie ring action of the Klein four-group on cyclic group:Z2, 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.

The cohomology group is isomorphic to elementary abelian group:E8.

Elements

Summary

Template:Second cohomology group for trivial Lie ring action facts to check against

Each element of the second cohomology group corresponds to a Lie ring extension with base ideal an abelian Lie ring isomorphic to cyclic group:Z2 in the center and the quotient Lie ring an abelian Lie ring whose additive group is isomorphic to Klein four-group. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each extension Lie ring is 2 \times 4 = 8.

Further, the minimum size of generating set of the extension group is at least equal to 2 (the minimum size of generating set of the quotient Klein four-group) and at most equal to 3 (the sum of the minimum size of generating set for the normal subgroup and quotient group).

Cohomology class type Number of cohomology classes Corresponding Lie ring extension Nilpotency class Isomorphism class of additive group
trivial 1 abelian Lie ring, additive group elementary abelian group:E8 1 elementary abelian group:E8
symmetric and nontrivial 3 abelian Lie ring, additive group direct product of Z4 and Z2 1 direct product of Z4 and Z2
non-symmetric 1 niltriangular matrix Lie ring:NT(3,2) 2 elementary abelian group:E8
non-symmetric 3 semidirect product of Z4 and Z2 as Lie rings 2 direct product of Z4 and Z2
Total (4 rows) 8 (= order of the cohomology group) -- -- --