Center 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:Z2 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 dihedral group:D8.
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The semidihedral group $SD_{16}$ (also denoted $QD_{16}$) is the semidihedral group (also called quasidihedral group) of order $16$. Specifically, it has the following presentation:

$G=SD_{16}:=\langle a,x\mid a^{8}=x^{2}=e,xax^{-1}=a^{3}\rangle$.

$G$ 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 $\langle a^{4}\rangle =\{a^{4},e\}$.

The quotient group is isomorphic to dihedral group:D8.

Cosets

The subgroup has order 2 and index 8, so it has 8 left cosets. It is a normal subgroup, so the left cosets coincide with the right cosets. The cosets are:

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

The quotient group is isomorphic to dihedral group:D8, and the multiplication table on cosets is given below. The row element is multiplied on the left and the column element is multiplied on the right.

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

Note that the multiplication table for the quotient group looks identical to that for center of dihedral group:D16, although the original groups differ (semidihedral group:SD16 versus dihedral group:D16).

Arithmetic functions

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

Subgroup-defining functions

Subgroup-defining function	What it means in general	Why it takes this value	GAP verification (set G := SmallGroup(16,8); H := Center(G))
center	elements that commute with every group element	No element of the form $a^{k}x$ can be central since it does not commute with $a$. Among powers of $a$, an element is central iff it equals its inverse. The only two such elements are $e$ and $a^{4}$.	Definitional
third member of lower central series	commutator subgroup between whole group and its derived subgroup, i.e., $[[G,G],G]$	Derived subgroup is $\langle a^{2}\rangle$, and taking the commutator again gives $\langle a^{4}\rangle$.	H = CommutatorSubgroup(G,DerivedSubgroup(G)); using DerivedSubgroup and CommutatorSubgroup.
socle	join of all minimal normal subgroups	In fact, it is the unique minimal normal subgroup -- the group is a monolithic group.	H = Socle(G); using Socle
second agemo subgroup	subgroup generated by elements of the form $g^{p^{2}}$, here $p=2$ so subgroup generated by all elements of the form $g^{4}$	Every fourth power is either $e$ or $a^{4}$.	H = Agemo(G,2,2); using Agemo

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	center is normal
characteristic subgroup	invariant under all automorphisms	Yes	center is characteristic
fully invariant subgroup	invariant under all endomorphisms	Yes	lower central series members are fully invariant, agemo subgroups are fully invariant
verbal subgroup	generated by set of words	Yes	Lower central series members are verbal, agemo subgroups are verbal
normal-isomorph-free subgroup	no other isomorphic normal subgroup	Yes
isomorph-free subgroup, isomorph-containing subgroup	No other isomorphic subgroups	No	There are other subgroups of order two, such as $\{x,e\}$.
isomorph-normal subgroup	Every isomorphic subgroup is normal	No	There are other subgroups of order two that are not normal: $\langle x\rangle$ etc.
homomorph-containing subgroup	contains all homomorphic images	No	There are other subgroups of order two.
1-endomorphism-invariant subgroup	invariant under all 1-endomorphisms of the group	Yes	It is precisely the set of fourth powers, which must therefore go to squares under 1-endomorphisms
1-automorphism-invariant subgroup	invariant under all 1-automorphisms of the group	Yes	Follows from being 1-endomorphism-invariant.
quasiautomorphism-invariant subgroup	invariant under all quasiautomorphisms	Yes	Follows from being 1-automorphism-invariant