Derived subgroup 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:Z4 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 Klein four-group.
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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 subgroup $H$ of interest is the subgroup $\langle a^2 \rangle$. It is cyclic of order 4 and is given by:

$H := \{ e, a^2, a^4, a^6 \}$

The quotient group is a Klein four-group.

Cosets

The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets:

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

Arithmetic functions

Function	Value	Explanation
order of whole group	16
order of subgroup	4
index of subgroup	4
size of conjugacy class (=index of normalizer)	1
number of conjugacy classes in automorphism class	1

Subgroup-defining functions

Subgroup-defining function	Meaning in general	Why it takes this value
derived subgroup	subgroup generated by all commutators	The commutators are precisely the elements of this subgroup. For instance, $xax^{-1}a^{-1} = a^{-2}$.
first agemo subgroup	subgroup generated by all $p^{th}$ powers. Here, $p = 2$, so subgroup generated by squares
Frattini subgroup	intersection of all maximal subgroups
Jacobson radical	intersection of all maximal normal subgroups

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	derived subgroup is normal
characteristic subgroup	invariant under all automorphisms	Yes	derived subgroup is characteristic
fully invariant subgroup	invariant under all endomorphisms	Yes	derived subgroup is fully invariant, agemo subgroups are fully invariant
isomorph-free subgroup	no other isomorphic subgroup	Yes
verbal subgroup	generated by set of words	Yes	derived subgroup is verbal, agemo subgroups are verbal
homomorph-containing subgroup	contains every homomorphic image	No	The subgroup $\{ x, e \}$ is an image of this subgroup but is not contained in it.

GAP implementation

The group and subgroup can be constructed using GAP's SmallGroup and DerivedSubgroup functions:

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

The GAP display looks as follows:

gap> G := DihedralGroup(16); H := DerivedSubgroup(G);
<pc group of size 16 with 4 generators>
Group([ f3, f4 ])

Here is a GAP implementation to verify some of the assertions made in this page:[SHOW MORE]

gap> StructureDescription(G/H);
"C2 x C2"
gap> Order(G);
16
gap> Order(H);
4
gap> Index(G,H);
4
gap> IsNormal(G,H);
true
gap> H = FrattiniSubgroup(G);
true
gap> H = Agemo(G,2,1);
true
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
true