Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) nontrivial semidirect product of Z4 and Z4 (see subgroup structure of nontrivial semidirect product of Z4 and Z4).
The subgroup is a normal subgroup and the quotient group is isomorphic to dihedral group:D8.
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Consider the group (here denotes the identity element):
This is a group of order 16 with elements:
We are interested in the subgroup:
This is a subgroup generated by the square which is not a commutator (the only commutators are and ).
The quotient group is isomorphic to dihedral group:D8. We can see this by noting that the presentation of is the same as that of but now with the additional constraint that is the identity element. This is precisely the presentation of the dihedral group of order 8.
The subgroup is a normal subgroup, so the left and right cosets coincide. The 8 cosets are:
|order of whole group||16|
|order of subgroup||2|
|size of conjugacy class||1|
|number of conjugacy classes in automorphism class||1|
Invariance under automorphisms and endomorphisms
|normal subgroup||invariant under inner automorphisms||Yes||central implies normal|
|characteristic subgroup||invariant under all automorphisms||Yes||it is the only subgroup generated by a square that is not a commutator|
|fully invariant subgroup||invariant under all endomorphisms||No||The endomorphism sends this subgroup to .|
|isomorph-characteristic subgroup||Every isomorphic subgroup is characteristic||Yes||The two other isomorphic subgroups are both characteristic, see central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 and derived subgroup of nontrivial semidirect product of Z4 and Z4.|
|characteristic-isomorph-free subgroup||no other isomorphic characteristic subgroup||No||and are isomorphic characteristic subgroups. Hence also not a isomorph-free subgroup, isomorph-containing subgroup, or normal-isomorph-free subgroup.|
|isomorph-normal subgroup||Every isomorphic subgroup is normal||Yes||Follows from being isomorph-characteristic.|
|central subgroup||invariant under inner automorphisms||Yes||The center is .|
|central factor||product with centralizer is whole group||Yes||central implies central factor|
The group and subgroup pair can be constructed as follows:
G := SmallGroup(16,4); H := Group(Difference(Set(List(G,x -> x^2)),Set(DerivedSubgroup(G))));
Here is the GAP display for this:
gap> G := SmallGroup(16,4); H := Group(Difference(Set(List(G,x -> x^2)),Set(DerivedSubgroup(G)))); <pc group of size 16 with 4 generators> <pc group with 1 generators>
Here is GAP code to verify some of the assertions in this page:
gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> StructureDescription(H); "C2" gap> StructureDescription(G/H); "D8" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false