Center of dihedral group: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) cyclic group:Z2 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 dihedral group:D8.
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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 subgroup $H$ of interest is the subgroup $\langle a^{4}\rangle =\{e,a^{4}\}$.

The quotient group is isomorphic to dihedral group:D8.

The subgroup is isomorphic to cyclic group:Z2 and the multiplication table is given below. Note that the group is an abelian group, so we do not need to worry about left/right issues:

Element/element	$e$	$a^{4}$
$e$	$e$	$a^{4}$
$a^{4}$	$a^{4}$	$e$

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

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 := DihedralGroup(16); H := Center(G);) -- see more at #GAP implementation
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 CommutatorSubgroup and DerivedSubgroup
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	GAP verification (set G := DihedralGroup(16);H:= Center(G);) -- see more at #GAP implementation
normal subgroup	invariant under inner automorphisms	Yes	center is normal	IsNormal(G,H); using IsNormal
characteristic subgroup	invariant under all automorphisms	Yes	center is characteristic	IsCharacteristicSubgroup(G,H); using IsCharacteristicSubgroup
fully invariant subgroup	invariant under all endomorphisms	Yes	lower central series members are fully invariant, agemo subgroups are fully invariant	IsFullinvariant(G,H); using IsFullinvariant
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

GAP implementation

The group and subgroup can be constructed using GAP's DihedralGroup and Center functions as follows:

G := DihedralGroup(16); H := Center(G);

The GAP display looks as follows:

gap> G := DihedralGroup(16); H := Center(G);
<pc group of size 16 with 4 generators>
Group([ f4 ])

Here is a GAP implementation to verify some of the assertions made on this page:

gap> Order(G);
16
gap> Order(H);
2
gap> Index(G,H);
8
gap> StructureDescription(G/H);
"D8"
gap> H = CommutatorSubgroup(G,DerivedSubgroup(G));
true
gap> H = Socle(G);
true
gap> H = Agemo(G,2,2);
true
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
true