# 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.