Second cohomology group for trivial group action of Z2 on Z2

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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

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.,

\! 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.

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 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.

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

Homomorphisms to and from other cohomology groups

Homomorphisms on A

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

The unique injective homomorphism from A = \mathbb{Z}_2 to \mathbb{Z}_4 induces a homomorphism:

H^2(G;A) \to H^2(G;\mathbb{Z}_4)

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 H^2(G;\mathbb{Z}_2) 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 \mathbb{Z}_2 is the double of something in the target group) is trivial on second cohomology.

The unique surjective homomorphism from \mathbb{Z}_4 to \mathbb{Z}_2 induces a homomorphism:

H^2(G;\mathbb{Z}_4) \to H^2(G;A)

This map is an isomorphism of groups.

Homomorphisms on G

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 \mathbb{Z}_2 and the base normal subgroup has center equal to \mathbb{Z}_2. In this case, the set of congruence classes of extensions is classified by H^2(\mathbb{Z}_2,\mathbb{Z}_2). 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.

Group with center \mathbb{Z}_2 Order GAP ID (second part) Information on extensions Trivial extension (direct product) GAP ID (second part) Nontrivial extension GAP ID (second part) Is the nontrivial extension semidirect?
cyclic group:Z2 2 1 current page Klein four-group 2 cyclic group:Z4 1 No
dihedral group:D8 8 3 extensions for trivial outer action of Z2 on D8 direct product of D8 and Z2 11 central product of D8 and Z4 13 Yes
quaternion group 8 4 extensions for trivial outer action of Z2 on Q8 direct product of Q8 and Z2 12 central product of D8 and Z4 13 Yes
dihedral group:D16 16 7 extensions for trivial outer action of Z2 on D16 direct product of D16 and Z2 39 central product of D16 and Z4 42 Yes
semidihedral group:SD16 16 8 extensions for trivial outer action of Z2 on SD16 direct product of SD16 and Z2 40 central product of D16 and Z4 42 Yes
generalized quaternion group:Q16 16 9 extensions for trivial outer action of Z2 on Q16 direct product of Q16 and Z2 41 central product of D16 and Z4 42 Yes


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 ] ]