Non-central Z4 in M16
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:Z4 and the group is (up to isomorphism) M16 (see subgroup structure of M16).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z4.
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We consider the group:
with denoting the identity element.
This is a group of order 16, with elements:
We are interested in the subgroup:
Here is the multiplication table for . Note that is an abelian group so we don't have to worry about left/right issues:
Below is the multiplication for . It is a cyclic group of order 4 with generator the coset . Since it is an abelian group, we do not need to worry about left/right convention for the multiplication table:
Invariance under automorphisms and endomorphisms
|normal subgroup||invariant under inner automorphisms||Yes||Follows from being the only non-central cyclic group of order four.|
|characteristic subgroup||invariant under all automorphisms||Yes||Follows from being the only non-central cyclic group of order four.|
|fully invariant subgroup||invariant under all endomorphisms||No||The endomorphism (which is a retraction onto ) does not preserve this subgroup.|
|isomorph-free subgroup||no other isomorphic subgroup||No||center of M16 is another isomorphic subgroup.|
|isomorph-characteristic subgroup||every isomorphic subgroup is characteristic||Yes||The only isomorphic subgroups are the subgroup itself and center of M16.|
The group and subgroup pair can be constructed as follows:
G := SmallGroup(16,6); H := Filtered(NormalSubgroups(G),x -> Order(x) = 4 and IsCyclic(x) and not(x = Center(G)));
Here is the GAP display:
gap> G := SmallGroup(16,6); H := Filtered(NormalSubgroups(G),x -> Order(x) = 4 and IsCyclic(x) and not(x = Center(G))); <pc group of size 16 with 4 generators> Group([ f2*f3, f4 ])
Here is GAP code to verify some of the assertions in this page:
gap> Order(G); 16 gap> Order(H); 4 gap> Index(G,H); 4 gap> StructureDescription(H); "C4" gap> StructureDescription(G/H); "C4" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false