Central subgroup generated by a non-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 quaternion group.
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Consider the group:
This is a group of order 16 with elements:
We are interested in the subgroup:
The quotient group is isomorphic to quaternion group.
|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||The non-identity element is the unique element in the center that is not a square.|
|fully invariant subgroup||invariant under all endomorphisms||No||The endomorphism sends this subgroup to , the derived subgroup of nontrivial semidirect product of Z4 and Z4.|
The group and subgroup pair can be constructed using GAP as follows:
G := SmallGroup(16,4); H := Group(Difference(Set(Center(G)),Set(List(G,x -> x^2))));
The GAP display looks as below:
gap> G := SmallGroup(16,4); H := Group(Difference(Set(Center(G)),Set(List(G,x -> x^2)))); <pc group of size 16 with 4 generators> <pc group with 1 generators>
Here is some GAP code verifying some of the assertions on this page:
gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> StructureDescription(H); "C2" gap> StructureDescription(G/H); "Q8" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false