Center of direct product of D8 and Z2

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) direct product of D8 and Z2 (see subgroup structure of direct product of D8 and Z2).
The subgroup is a normal subgroup and the quotient group is isomorphic to Klein four-group.
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Definition

The group $G$ is direct product of D8 and Z2, given as follows:

$G:=\langle a,x,y\mid a^{4}=x^{2}=y^{2}=e,xax^{-1}=a^{-1},ay=ya,xy=yx\rangle$

The subgroup $\langle a,x\rangle$ is the first direct factor (dihedral group:D8) and the subgroup $\langle y\rangle$ is the second direct factor (cyclic group:Z2).

$G$ has 16 elements:

$\!e,a,a^{2},a^{3},x,ax,a^{2}x,a^{3}x,y,ay,a^{2}y,a^{3}y,xy,axy,a^{2}xy,a^{3}xy$

The subgroup we are interested in is:

$H:=\langle a^{2},y\rangle =\{e,a^{2},y,a^{2}y\}$

$H$ is the center of $G$ and is isomorphic to the Klein four-group.

Cosets

$H$ is a normal subgroup of $G$ and it has four cosets:

$\!\{e,a^{2},y,a^{2}y\},\{a,a^{3},ay,a^{3}y\},\{x,a^{2}x,xy,a^{2}xy\},\{ax,a^{3}x,axy,a^{3}xy\}$

The quotient group is also isomorphic to the Klein four-group.

Arithmetic functions

Function	Value	Explanation
order of whole group	16
order of subgroup	4
index of subgroup	4
size of conjugacy class of subgroup	1
number of conjugacy classes in automorphism class	1
size of automorphism class of subgroup	1

Subgroup-defining functions

Subgroup-defining function	Meaning in general	Why it takes this value	GAP verification (set `G := DirectProduct(DihedralGroup(8),CyclicGroup(2));H := Center(G)`)
center	precisely the elements that commute with every element	center of direct product is direct product of centers, and the center of dihedral group:D8 is $\langle a^{2}\rangle$	`H = Center(G);` using Center
socle	join of all minimal normal subgroups. For a group of prime power order, same as $\Omega _{1}(Z(G))$ (because minimal normal implies central in nilpotent).	Center is already elementary abelian, so it equals the socle.	`H = Center(G);` using Center

Subgroup properties

Invariance under automorphisms and endomorphisms: basic properties

Property	Satisfied?	Explanation	GAP verification (set `G := DirectProduct(DihedralGroup(8),CyclicGroup(2));H := Center(G)`) -- see #GAP implementation
normal subgroup	Yes	center is normal	`IsNormal(G,H);` using IsNormal
characteristic subgroup	Yes	center is characteristic	`IsCharacteristicSubgroup(G,H);` using IsCharacteristicSubgroup
fully invariant subgroup	No	Consider the endomorphism $a\mapsto a^{2},x\mapsto e,y\mapsto x$ . This endomorphism does not preserve the subgroup.	`IsFullinvariant(G,H);` using IsFullinvariant

Word expression related properties and corollaries about invariance under automorphisms and endomorphisms

Property	Meaning in general	Satisfied?	Explanation
marginal subgroup	(complicated)	Yes	center is marginal
verbal subgroup	generated by set of words	No	verbal implies fully invariant, the subgroup is not fully invariant
finite direct power-closed characteristic subgroup	any finite direct power is characteristic in the corresponding direct power of the whole group	Yes	center is finite direct power-closed characteristic (also via marginal)