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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Definition

Consider the group:

$G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$.

This is a group of order 16 with elements:

$\! \{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}$

We are interested in the subgroup:

$\! H = \{ e, x^2, y^2, x^2y^2 \}$

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 $H$ is given as follows:

Element/element $e$ $x^2$ $y^2$ $x^2y^2$
$e$ $e$ $x^2$ $y^2$ $x^2y^2$
$x^2$ $x^2$ $e$ $x^2y^2$ $y^2$
$y^2$ $y^2$ $x^2y^2$ $e$ $x^2$
$x^2y^2$ $x^2y^2$ $y^2$ $x^2$ $e$

Cosets

$H$ is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets, because the index of $H$ in $G$ is $16/4 = 4$. These are:

$\{ e, x^2, y^2, x^2y^2 \}, \{ x, x^3, xy^2, x^3y^2 \}, \{ y, x^2y, y^3, x^2y^3 \}, \{ xy, x^3y, xy^3, x^3y^3 \}$

The quotient group is isomorphic to a Klein four-group, and the multiplication table is given as follows:

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 $\Omega_1(Z(G))$ The minimal normal subgroups are $\{ e, x^2 \}$ (see here), $\{ e, y^2 \}$ (see here), $\{ e, x^2y^2 \}$ (see here). H = Socle(G); using Socle
first omega subgroup generated by the set of all elements of order $p$ (here $p = 2$) The elements of order 2 are precisely $x^2, y^2, x^2y^2$.
[[arises as subgroup-defining function::agemo subgroups of group of prime power order|first agemo subgroup generated by the set of all $p^{th}$ powers (here $p = 2$, so squares) The squares are precisely $e, x^2, y^2$ (note $x^2y^2$ 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