Center of 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) Klein four-group 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 Klein four-group.
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Contents
Definition
Consider the group:
.
This is a group of order 16 with elements:
We are interested in the subgroup:
This is the center. In particular, it is a normal subgroup isomorphic to the Klein four-group and the quotient group is also isomorphic to the Klein four-group.
The multiplication table for is given as follows:
Element/element | ||||
---|---|---|---|---|
Cosets
is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets, because the index of in is . These are:
The quotient group is isomorphic to a Klein four-group, and the multiplication table is given as follows:
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value | GAP verification (set G := SmallGroup(16,4); H := Center(G); before starting) |
---|---|---|---|
center | elements that commute with every group element | Follows from multiplication table | Definitional |
socle | join of all minimal normal subgroups, equivalently | The minimal normal subgroups are (see here), (see here), (see here). | H = Socle(G); using Socle |
first omega subgroup | generated by the set of all elements of order (here ) | The elements of order 2 are precisely . | |
[[arises as subgroup-defining function::agemo subgroups of group of prime power order|first agemo subgroup | generated by the set of all powers (here , so squares) | The squares are precisely (note is not a square, but is in the subgroup generated). | H = Agemo(G,2,1); using Agemo |
Subgroup properties
Invariance under automorphisms and endomorphisms
Property | Meaning | Satisfied? | Explanation | GAP verification (set G := SmallGroup(16,4); H := Center(G); first) -- see #GAP implementation |
---|---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | center is normal | IsNormal(G,H); using IsNormal |
characteristic subgroup | invariant under all automorphisms | Yes | center is characteristic | IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup |
fully invariant subgroup | invariant under all endomorphisms | Yes | omega subgroups are fully invariant, agemo subgroups are fully invariant | IsFullinvariant(G,H); using IsFullinvariant |
verbal subgroup | generated by set of words | Yes | agemo subgroups are verbal | |
isomorph-free subgroup | no other isomorphic subgroup | Yes | precisely the set of elements of order two (plus the identity element). | |
homomorph-containing subgroup | contains any homomorphic image | Yes | precisely the set of elements of order two (plus the identity element). |
GAP implementation
The group and subgroup pair can be constructed using GAP's SmallGroup and Center functions:
G := SmallGroup(16,4); H := Center(G);
Here is the GAP display:
gap> G := SmallGroup(16,4); H := Center(G); <pc group of size 16 with 4 generators> Group([ f3, f4 ])
Here is some GAP code to verify the assertions on this page:
gap> Order(G); 16 gap> Order(H); 4 gap> Index(G,H); 4 gap> StructureDescription(H); "C2 x C2" gap> StructureDescription(G/H); "C2 x C2" gap> H = Socle(G); true gap> H = Agemo(G,2,1); true gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true