Non-normal subgroups of dihedral group:D8

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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:D8 (see subgroup structure of dihedral group:D8).
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Definition

Suppose G is the dihedral group of order eight (degree four) given by the presentation below, where e denotes the identity element of G:

G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle.

Then, we are interested in the following four subgroups:

A_0 := \langle x \rangle = \{ x, e \}, A_1 := \langle ax \rangle = \{ ax, e \}, A_2 := \langle a^2x \rangle = \{ a^2x, e \}, A_3 = \langle a^3x \rangle = \{ a^3x, e \}.

A_0 and A_2 are conjugate subgroups (via a, for instance). A_1 and A_3 are conjugate subgroups (via a, for instance). A_0 and A_1 are not conjugate but are related by an outer automorphism that fixes a and sends x to ax. Thus, all four subgroups are automorphic subgroups. These are the only non-normal subgroups of G and they are all 2-subnormal subgroups.

Arithmetic functions

Function Value Explanation
order of whole group 8
order of subgroup 2
index 2
size of conjugacy class 2
number of conjugacy classes in automorphism class 2
size of automorphism class 2
subnormal depth 2
hypernormalized depth 2

Effect of subgroup operators

Specific values (in the second column) are for A_0 = \langle x \rangle.

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer \langle a^2, x \rangle Klein four-subgroups of dihedral group:D8 Klein four-group
centralizer \langle a^2,x \rangle Klein four-subgroups of dihedral group:D8 Klein four-group
normal core \{ e \} -- trivial group
normal closure \langle a^2,x \rangle Klein four-subgroups of dihedral group:D8 Klein four-group
characteristic core \{ e \} -- trivial group
characteristic closure G, i.e., \langle a,x \rangle -- dihedral group:D8

Related subgroups

Intermediate subgroups

We use A_0 = \langle x\rangle here.

Value of intermediate subgroup (descriptive) Isomorphism class of intermediate subgroup Small subgroup in intermediate subgroup Intermediate subgroup in big group
\langle a^2, x \rangle Klein four-group Z2 in V4 Klein four-subgroups of dihedral group:D8

Subgroup properties

Invariance under automorphisms and endomorphisms

Suppose c_a and c_x denote conjugation by a and x respectively. Let \sigma denote the automorphism that sends a to a^3 and x to ax. Then, \langle c_a, c_x\rangle is the inner automorphism group and \langle c_a, c_x, \sigma \rangle is the automorphism group.

The automorphism c_x fixes A_0 and A_2 while interchanging A_1 and A_3. The automorphism c_a interchanges A_0 and A_2 while also interchanging A_1 and A_3. The automorphism c_{ax} = c_a \circ c_x fixes A_1 and A_3 while interchanging A_0 and A_2. The automorphism \sigma interchanges A_0 and A_1 and also interchanges A_2 and A_3.

Property Meaning Satisfied? Explanation Comment
normal subgroup invariant under inner automorphisms No See above description of conjugation automorphisms that permute the subgroups
coprime automorphism-invariant subgroup invariant under automorphisms of coprime order to group Yes there are no nontrivial automorphisms of coprime order
cofactorial automorphism-invariant subgroup invariant under all automorphisms whose order has prime factors only among those of the group No follows from not being normal
2-subnormal subgroup normal subgroup of normal subgroup Yes normal inside Klein four-subgroups of dihedral group:D8 (of the form \langle a^2,x \rangle and \langle a^2,ax\rangle) that are normal in the whole group.
subnormal subgroup Yes follows from being 2-subnormal, also from being subgroup of nilpotent group.