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:

$\!\{e,a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},x,ax,a^{2}x,a^{3}x,a^{4}x,a^{5}x,a^{6}x,a^{7}x\}$

The subgroups $H_{1}$ and $H_{2}$ of interest are:

$\!H_{1}=\{e,a^{2},a^{4},a^{6},x,a^{2}x,a^{4}x,a^{6}x\},\qquad H_{2}=\{e,a^{2},a^{4},a^{6},ax,a^{3}x,a^{5}x,a^{7}x\}$

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:

$\!H_{1}=\{e,a^{2},a^{4},a^{6},x,a^{2}x,a^{4}x,a^{6}x\},G\setminus H_{1}=\{a,a^{3},a^{5},a^{7},ax,a^{3}x,a^{5}x,a^{7}x\}$

The cosets of $H_{2}$ are:

$H_{2}=\{e,a^{2},a^{4},a^{6},ax,a^{3}x,a^{5}x,a^{7}x\},G\setminus H_{2}=\{a,a^{3},a^{5},a^{7},x,a^{2}x,a^{4}x,a^{6}x\}$

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.