Cyclic maximal subgroup of semidihedral group:SD16

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) cyclic group:Z8 and the group is (up to isomorphism) semidihedral group:SD16 (see subgroup structure of semidihedral group:SD16).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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Here, $G$ is the semidihedral group:SD16, the semidihedral group of order sixteen (and hence, degree eight). We use here the presentation:

$G:=\langle a,x\mid a^{8}=x^{2}=e,xax=a^{3}\rangle$

$G$ 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 subgroup $H$ of interest is the subgroup $\langle a\rangle$. It is cyclic of order 8 and is given by:

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

Cosets

The subgroup has index two and is hence normal (since index two implies normal). Its left cosets coincide with its right cosets, and there are two cosets:

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

Complements

COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)

There are four possible permutable complements to $H$ in $G$, all of them automorphic to each other:

$\!\{e,x\},\{e,a^{2}x\},\{e,a^{4}x\},\{e,a^{6}x\}$

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
complemented normal subgroup	normal subgroup with permutable complement	Yes	see above
permutably complemented subgroup	subgroup with permutable complement	Yes	(via complemented normal)
lattice-complemented subgroup	subgroup with lattice complement	Yes	(via permutably complemented)
retract	has a normal complement	No
direct factor	normal subgroup with normal complement	No

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	1

Effect of subgroup operators

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	whole group	--	semidihedral group:D16
centralizer	$\langle a\rangle$ -- the subgroup itself	current page	cyclic group:Z8
normal core	the subgroup itself	current page	cyclic group:Z8
normal closure	the subgroup itself	current page	cyclic group:Z8
characteristic core	the subgroup itself	current page	cyclic group:Z8
characteristic closure	the subgroup itself	current page	cyclic group:Z8

Subgroup-defining functions

Subgroup-defining function	Meaning in general	Why it takes this value
centralizer of derived subgroup	centralizer of the derived subgroup (the commutator of the group with itself)	The derived subgroup is $\langle a^{2}\rangle$, which is centralized by all of $\langle a\rangle$ and by nothing outside it.

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes
characteristic subgroup	invariant under all automorphisms	Yes	On account of being the centralizer of derived subgroup, also on account of being an isomorph-free subgroup.
fully invariant subgroup	invariant under all endomorphisms	No	not invariant under the retraction with kernel $\langle a^{2},ax\rangle$ and image $\{e,x\}$.
isomorph-free subgroup	no other isomorphic subgroup	Yes