Q8 in central product of D8 and Z4

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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) quaternion group and the group is (up to isomorphism) central product of D8 and Z4 (see subgroup structure of central product of D8 and Z4).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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The group G is central product of D8 and Z4:

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

The group is a central product of dihedral group:D8 and cyclic group:Z4, and is also a central product of the quaternion group and cyclic group:Z4. The group 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

We are interested in the subgroup:

H := \langle a, xy \rangle = \{ e, a, a^2, a^3, xy, axy, a^2xy, a^3xy \}

This group is isomorphic to the quaternion group, with e \mapsto 1, a^2 \mapsto -1.

Cosets

The subgroup has index two and index two implies normal, so it is a normal subgroup and thus its left cosets coincide with its right cosets. The two cosets are:

H = \{ e, a, a^2, a^3, xy, axy, a^2xy, a^3xy \}, G \setminus H = \{ x, ax, a^2x, a^3x, y, ay, a^2y, a^3y \}

Arithmetic functions

Function Value Explanation
order of whole group 16
order of subgroup 8
index of subgroup 2
size of conjugacy class of subgroup (= index of normalizer) 1
number of conjugacy classes in automorphism class of subgroup 1
size of automorphism class of subgroup 1

Subgroup properties

Invariance under automorphisms and endomorphisms: basic properties

Property Meaning Satisfied? Explanation
normal subgroup invariant under all inner automorphisms Yes index two implies normal
characteristic subgroup invariant under all automorphisms Yes Follows from being isomorph-free (see below)
fully invariant subgroup invariant under all endomorphisms No

Properties based on resemblance and invariance

Property Meaning Satisfied? Explanation
isomorph-free subgroup no other isomorphic subgroup Yes
isomorph-containing subgroup contains every isomorphic subgroup (same as isomorph-free for finite groups) Yes
homomorph-containing subgroup contains every homomorphic image No

Properties related to centrality and transitivity of normality

Property Meaning Satisfied? Explanation
cocentral subgroup product with center of whole group is whole group Yes
central factor product with centralizer is whole group Yes
central subgroup contained in the center No
transitively normal subgroup every normal subgroup of the subgroup is normal in the whole group Yes central factor implies transitively normal
hereditarily normal subgroup every subgroup of the subgroup is normal in the whole group Yes Follows from the subgroup being transitively normal and a Dedekind group -- all its subgroups are normal. See subgroup structure of quaternion group.