# Second cohomology group for trivial group action of D8 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 dihedral group:D8 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and dihedral group:D8 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.

The value of this cohomology group is elementary abelian group:E8.

Get more specific information about dihedral group:D8 |Get more specific information about cyclic group:Z2|View other constructions whose value is elementary abelian group:E8

## Contents

## Definition

This article is about the second cohomology group for trivial group action where the acting group is dihedral group:D8 (the dihedral group of order 8 and degree 4) and the base group is cyclic group:Z2 (the cyclic group of order 2). In other words, we are interested in:

where and .

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

## Computation of cohomology group

The cohomology group can be computed as an abstract group using group cohomology of dihedral group:D8.

## Elements

### Summary

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.

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 3) | Subgroup information on base in whole group |
---|---|---|---|---|---|---|---|---|---|---|

trivial | 1 | direct product of D8 and Z2 | 11 | No | No | 2 | 2 | 3 | ||

nontrivial and symmetric | 1 | nontrivial semidirect product of Z4 and Z4 | 4 | No | Yes | 2 | 2 | 2 | Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4 | |

nontrivial and symmetric | 2 | SmallGroup(16,3) | 3 | No | No | 2 | 2 | 2 | ||

non-symmetric | 1 | dihedral group:D16 | 7 | Yes | Yes | 3 | 2 | 2 | center of dihedral group:D16 | |

non-symmetric | 1 | generalized quaternion group:Q16 | 9 | Yes | Yes | 3 | 2 | 2 | center of generalized quaternion group:Q16 | |

non-symmetric | 2 | semidihedral group:SD16 | 8 | Yes | Yes | 3 | 2 | 2 | center of semidihedral group:SD16 | |

Total (6 rows) | 8 (equals order of the cohomology group) | -- | -- | -- | -- | -- | -- | -- | -- | -- |

Note that all these extensions are central extensions with the base normal subgroup isomorphic to cyclic group:Z2 and the quotient group isomorphic to dihedral group:D8. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each of the extension groups is .

Some, but not all, of the extensions are stem extensions. Since cyclic group:Z2 is in fact *also* the Schur multiplier of dihedral group:D8, the stem extensions here are precisely the Schur covering groups of dihedral group:D8.

The minimum size of generating set of the extension group is *at least* equal to 2 (which is the minimum size of generating set of the quotient group) and *at most* equal to 3 (which is the sum of the minimum size of generating set of the normal subgroup and the quotient group). See minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group and minimum size of generating set of quotient group is at most minimum size of generating set of group.

The nilpotency class is at least 2 and at most 3 in all cases. It is at least 2 because the quotient dihedral group:D8 has nilpotency class two. It is at most 3 because the sum of the nilpotency class of the normal subgroup and quotient group is 3, and the extension is a central extension. The derived length is always exactly 2 because nilpotency class 2 or 3 forces derived length exactly 2, using derived length is logarithmically bounded by nilpotency class.

### Explicit description and relation with power-commutator presentation

`Further information: power-commutator presentation, presentations for groups of order 16#Power-commutator presentations`

Consider an extension group with central subgroup isomorphic to (cyclic group:Z2) and quotient group isomorphic to dihedral group:D8. Denote by elements of such that , has order two, and . Explicitly, we are considering as given by the presentation .

We pick elements of that live in the cosets corresponding to , with the additional condition that . Next, we pick as the non-identity element of .

We can now write a power-commutator presentation of in terms of . With the usual notation, we already have , and all the other coefficients involving 1-3 are zero. This leaves us to choose the values . However, we note that because in our specific choice, equals . Thus, we in fact have only four values to choose: .

Further, it turns out that of these four values, we must have under these conditions. To see this, note that if . Because commutes with this can be rewritten as . Note that or , so its square is . The upshot is that . Inverting, we get .

The upshot of all this is that we can freely vary the parameters and the remaining parameters are fixed. This gives possibilities. Moreover, the coordinate-wise addition of these corresponds to addition in the cohomology group.

Note that the positions of dihedral group:D16 and semidihedral group:SD16 are sensitive to the commutator convention.

Corresponding extension group | Second part of GAP ID (order is 16) | Nilpotency class | |||
---|---|---|---|---|---|

0 | 0 | 0 | direct product of D8 and Z2 | 11 | 2 |

1 | 0 | 0 | nontrivial semidirect product of Z4 and Z4 | 4 | 2 |

0 | 1 | 0 | SmallGroup(16,3) | 3 | 2 |

1 | 1 | 0 | SmallGroup(16,3) | 3 | 2 |

0 | 0 | 1 | dihedral group:D16 | 7 | 3 |

1 | 0 | 1 | generalized quaternion group:Q16 | 9 | 3 |

0 | 1 | 1 | semidihedral group:SD16 | 8 | 3 |

1 | 1 | 1 | semidihedral group:SD16 | 8 | 3 |

## Group actions

### Under the action of the automorphism group of the acting group

By pre-composition, the automorphism group of dihedral group:D8, which is itself isomorphic to dihedral group:D8, acts on the second cohomology group. This action is transitive on all the extensions in each cohomology class type. In particular, the four fixed points are the cohomology classes corresponding to the extension groups direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, dihedral group:D16 and generalized quaternion group:Q16, and the two orbits of size two correspond to the extensions SmallGroup(16,3) and semidihedral group:SD16.

## Subgroups of interest

## Direct sum decomposition

For background information, see formula for second cohomology group for trivial group action in terms of second homology group and abelianization

### General background

We know from the general theory that there is a natural short exact sequence:

Here:

- The group corresponds to the subgroup of cohomology classes represented by symmetric 2-cocycles. is the abelianization of and its image comprises those extensions where the restricted extension of the derived subgroup on is trivial
*and*the corresponding extension of the quotient group is abelian. - The group is the second cohomology group up to isoclinism: Explicitly, is the Schur multiplier of .

We also know, again from the general theory, that the short exact sequence above splits, i.e., the image of in has a complement inside . However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

### In this case

For this choice of and , is isomorphic to the Klein four-group. The corresponding group is also a Klein four-group.

The Schur multiplier is cyclic group:Z2, hence is also isomorphic to cyclic group:Z2.

The image of in comprises the four non-stem extensions. It has two cosets in the whole second cohomology group. To split the short exact sequence in an automorphism-invariant fashion, we could pick as our complement either dihedral group:D16 or generalized quaternion group:Q16. The two possibilities are shown below:

Using dihedral group:D16 as the choice of complement, where the rows represent the cosets of the image of , and the columns represent the cosets of the complement:

Using generalized quaternion group:Q16 as the choice of complement, where the rows represent the cosets of the image of , and the columns represent the cosets of the complement:

### More on the mapping from

Below is an explicit description of the mapping from to , in terms of the original extension for Klein four-group (the abelianization of dihedral group:D8) on top of cyclic group:Z2 and what the new extension looks like for dihedral group:D8 on top of cyclic group:Z2.

Cohomology class type | Number of such cohomology classes | Element of as an extension | Image in |
---|---|---|---|

trivial | 1 | elementary abelian group:E8 | direct product of D8 and Z2 |

nontrivial | 1 | direct product of Z4 and Z2 | nontrivial semidirect product of Z4 and Z4 |

nontrivial | 2 | direct product of Z4 and Z2 | SmallGroup(16,3) |

Note that in , all the three nontrivial elements are in the same orbit under the natural action of the automorphism group of the Klein four-group. But they split into two orbits when we consider those automorphisms of the Klein four-group that are induced by automorphisms of the dihedral group:D8, which is why they give different outputs in .

## Homomorphisms to and from other cohomology groups

### Homomorphisms on

The unique injective homomorphism to induces a homomorphism:

The group on the right is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of D8 on Z4). However, the induced map above is *not* an isomorphism. Rather, it has kernel of order four precisely the image of in (see the direct sum decomposition section) and its image is a subgroup of order two in .

In terms of extensions, the map is interpreted as follows: it involves taking the central product of a given extension with cyclic group:Z4, identifying the base cyclic group:Z2 in the original extension with the in .

The map is given in the table below:

Input | Number of copies | Output = central product of input group with over identified central subgroup |
---|---|---|

direct product of D8 and Z2 | 1 | direct product of D8 and Z4 |

nontrivial semidirect product of Z4 and Z4 | 1 | direct product of D8 and Z4 |

SmallGroup(16,3) | 2 | direct product of D8 and Z4 |

dihedral group:D16 | 1 | central product of D16 and Z4 |

generalized quaternion group:Q16 | 1 | central product of D16 and Z4 |

semidihedral group:SD16 | 2 | central product of D16 and Z4 |

The unique surjective map from to induces a homomorphism:

The kernel of this map is the image of the preceding map and the image of this map is the kernel of the preceding map. The map is given in the table below:

## GAP implementation

### Construction of the cohomology group

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

gap> G := DihedralGroup(8);; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := <pc group of size 8 with 3 generators>, module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [ <an immutable 1x1 matrix over GF2>, <an immutable 1x1 matrix over GF2>, <an immutable 1x1 matrix over GF2> ] ), collector := rec( relators := [ [ 0 ], [ [ 2, 1, 3, 1 ], [ 3, 1 ] ], [ [ 3, 1 ], [ 3, 1 ], 0 ] ], orders := [ 2, 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1, 3, 1 ] ], estack := [ ], pstack := [ 3, 5 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ], list := [ 0, 0, 0 ], module := [ <an immutable 1x1 matrix over GF2>, <an immutable 1x 1 matrix over GF2>, <an immutable 1x1 matrix over GF2> ], mone := <an immutable 1x1 matrix over GF2>, mzero := <an immutable 1x1 matrix over GF2>, avoid := [ ], unavoidable := [ 1, 2, 3, 4, 5, 6 ] ), cohom := <linear mapping by matrix, <vector space of dimension 4 over GF( 2)> -> ( GF(2)^3 )>, presentation := rec( group := <free group on the generators [ f1, f2, f3 ]> , relators := [ f1^2, f1^-1*f2*f1*f3^-1*f2^-1, f2^2*f3^-1, f1^-1*f3*f1*f3\ ^-1, f2^-1*f3*f2*f3^-1, f3^2 ] ) )

### Construction of extensions

The extensions can be constructed using the additional command Extensions.

gap> G := DihedralGroup(8);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 16, 11 ], [ 16, 8 ], [ 16, 3 ], [ 16, 7 ], [ 16, 4 ], [ 16, 8 ], [ 16, 3 ], [ 16, 9 ] ]

### Under the action of the various automorphism groups

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

gap> G := ElementaryAbelianGroup(4);; 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); [ [ 16, 11 ], [ 16, 8 ], [ 16, 3 ], [ 16, 7 ], [ 16, 4 ], [ 16, 9 ] ]