D8 in D16

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) dihedral group:D8 and the group is (up to isomorphism) dihedral group:D16 (see subgroup structure of dihedral group:D16).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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Definition

Here, $G$ is the dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation: $G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle$ $G$ has 16 elements: $\! \{ e,a,a^2,a^3,a^4,a^5,a^6,a^7,x,ax,a^2x,a^3x,a^4x,a^5x,a^6x,a^7x \}$

The subgroups $H_1$ and $H_2$ of interest are: $\!H_1 = \{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}, \qquad H_2 = \{ e, a^2, a^4, a^6, ax, a^3x, a^5x, a^7x \}$

Both subgroups are normal but are related by the outer automorphism class of $a \mapsto a, x \mapsto ax$.

Cosets

Both subgroups has index two and are normal subgroup (See index two implies normal), so left cosets coincide with right cosets.

The cosets of $H_1$ are: $\! H_1 = \{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}, G \setminus H_1 = \{ a, a^3, a^5, a^7, ax, a^3x, a^5x, a^7x \}$

The cosets of $H_2$ are: $H_2 = \{ e, a^2, a^4, a^6, ax, a^3x, a^5x, a^7x \}, G \setminus H_2 = \{ a, a^3, a^5, a^7, x,a^2x,a^4x,a^6x \}$

Arithmetic functions

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

Subgroup properties

Invariance under automorphisms and endomorphisms

Property Meaning Satisfied? Explanation
normal subgroup invariant under inner automorphisms Yes index two implies normal
characteristic subgroup invariant under all automorphisms No The automorphism $a \mapsto a, x \mapsto ax$ interchanges $H_1$ and $H_2$.
fully invariant subgroup invariant under all endomorphisms No (follows from not being characteristic)
isomorph-free subgroup no other isomorphic subgroup No The two subgroups $H_1, H_2$ are isomorphic.
isomorph-automorphic subgroup all isomorphic subgroups are automorphic subgroups Yes $H_1,H_2$ are the only subgroups isomorphic to dihedral group:D8.
potentially fully invariant subgroup fully invariant subgroup inside a possibly bigger group No See D8 is not potentially fully invariant in D16.