Center of direct product of D8 and Z2

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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^2x, a^3x, y, ay, a^2y, a^3y, xy, axy, a^2xy, a^3xy

The subgroup we are interested in is:

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

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

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 GAP verification (set G := DirectProduct(DihedralGroup(8),CyclicGroup(2));H := Center(G)) -- see #GAP implementation
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)