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:

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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