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

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:Z2. The elements of this classify the group extensions with cyclic group:Z2 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:Z2

## Contents

## Description of the group

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

where and .

The cohomology group is isomorphic to cyclic group:Z2.

## Computation in terms of group cohomology

The group can be computed as an abstract group by using the group cohomology of cyclic group:Z2.

## Elements

FACTS TO CHECK AGAINST(second cohomology group for trivial group action):Background reading on relationship with extension groups: Group extension problemArithmetic 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 for the trivial group action congruence classes of central extensions with the specified subgroup and quotient group .

This descends to a correspondence:

Orbits for the group action of on 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.

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 | Corresponding group extension | Second part of GAP ID (order is 4) | Stem extension? | Base subgroup characteristic in whole group? | Nilpotency class of whole group | Derived length of whole group | Minimum size of generating set of whole group |
---|---|---|---|---|---|---|---|---|

trivial | 1 | Klein four-group | 2 | No | No | 1 | 1 | 2 |

nontrivial | 1 | cyclic group:Z4 | 1 | No | Yes | 1 | 1 | 1 |

Total (--) | 2 | -- | -- | -- | -- | -- | -- | -- |

## Subgroups of interest

Subgroup | Value | Corresponding group extensions for subgroup | GAP IDs (second part, order is 4) | Group extension groupings for each coset | GAP IDs (second part, order is 4) |
---|---|---|---|---|---|

IIP subgroup of second cohomology group for trivial group action | trivial subgroup | Klein four-group | 2 | (Klein four-group) and (cyclic group:Z4) | 2 (1 copy) and 1 (1 copy) |

cyclicity-preserving subgroup of second cohomology group for trivial group action | trivial subgroup | Klein four-group | 2 | (Klein four-group) and (cyclic group:Z4) | 2 (1 grouping) and 1 (1 grouping) |

subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions) | whole group | Klein four-group and cyclic group:Z4 | 2,1 | (Klein four-group and cyclic group:Z4) | (2 and 1) (1 grouping) |

## Generalizations

- Second cohomology group for trivial group action of group of prime order on group of prime order: This is a case where the prime number is 2.
- Second cohomology group for trivial group action of finite cyclic group on finite cyclic group: In this case, both finite cyclic groups are cyclic group:Z2.

## Homomorphisms to and from other cohomology groups

### Homomorphisms on

We note that the second cohomology group is covariant in the second coordinate.

The unique injective homomorphism from to induces a homomorphism:

The group on the right is also isomorphic to cyclic group:Z2 (see second cohomology group for trivial group action of Z2 on Z4). However, the map above is *not* an isomorphism. Rather, it is a trivial map, i.e., it sends everything in to the zero element on the right side. More general, any map that factors through a map of this form (i.e., where the image of the nonzero element of is the double of something in the target group) is trivial on second cohomology.

The unique surjective homomorphism from to induces a homomorphism:

This map is an isomorphism of groups.

### Homomorphisms on

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Application to other extensions

### Trivial outer action

We consider the cases where the action (by outer automorphisms) is trivial, the quotient is and the base normal subgroup has center equal to . In this case, the set of congruence classes of extensions is classified by . The external direct product corresponds to the zero element of the cohomology group. The central product with cyclic group:Z4 corresponds to the nontrivial extension.

## GAP implementation

### Construction of the cohomology group

The cohomology group can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF.

gap> G := CyclicGroup(2);; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := <pc group of size 2 with 1 generators>, module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [ <an immutable 1x1 matrix over GF2> ] ), collector := rec( relators := [ [ 0 ] ], orders := [ 2 ], wstack := [ [ 1, 1 ] ], estack := [ ], pstack := [ 3 ], cstack := [ 1 ], mstack := [ 0 ], list := [ 0 ], module := [ <an immutable 1x1 matrix over GF2> ], mone := <an immutable 1x1 matrix over GF2>, mzero := <an immutable 1x1 matrix over GF2>, avoid := [ ], unavoidable := [ 1 ] ), cohom := <linear mapping by matrix, <vector space of dimension 1 over GF( 2)> -> ( GF(2)^1 )>, presentation := rec( group := <free group on the generators [ f1 ]>, relators := [ f1^2 ] ) )

### Construction of extensions

The extensions can be constructed using the additional command Extensions.

gap> G := CyclicGroup(2);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 4, 2 ], [ 4, 1 ] ]

### Under the action of the various automorphism groups

This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := CyclicGroup(2);; gap> A := TrivialGModule(G,GF(2));; gap> A1 := AutomorphismGroup(G);; gap> A2 := GL(1,2);; gap> D := DirectProduct(A1,A2);; gap> P := CompatiblePairs(G,A,D);; gap> M := ExtensionRepresentatives(G,A,P);; gap> List(M,IdGroup); [ [ 4, 2 ], [ 4, 1 ] ]