Non-normal subgroups of dihedral group:D8
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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Suppose is the dihedral group of order eight (degree four) given by the presentation below, where denotes the identity element of :
Then, we are interested in the following four subgroups:
and are conjugate subgroups (via , for instance). and are conjugate subgroups (via , for instance). and are not conjugate but are related by an outer automorphism that fixes and sends to . Thus, all four subgroups are automorphic subgroups. These are the only non-normal subgroups of and they are all 2-subnormal subgroups.
|order of whole group||8|
|order of subgroup||2|
|size of conjugacy class||2|
|number of conjugacy classes in automorphism class||2|
|size of automorphism class||2|
Effect of subgroup operators
Specific values (in the second column) are for .
|Function||Value as subgroup (descriptive)||Value as subgroup (link)||Value as group|
|normalizer||Klein four-subgroups of dihedral group:D8||Klein four-group|
|centralizer||Klein four-subgroups of dihedral group:D8||Klein four-group|
|normal core||--||trivial group|
|normal closure||Klein four-subgroups of dihedral group:D8||Klein four-group|
|characteristic core||--||trivial group|
|characteristic closure||, i.e.,||--||dihedral group:D8|
We use here.
|Value of intermediate subgroup (descriptive)||Isomorphism class of intermediate subgroup||Small subgroup in intermediate subgroup||Intermediate subgroup in big group|
|Klein four-group||Z2 in V4||Klein four-subgroups of dihedral group:D8|
Invariance under automorphisms and endomorphisms
Suppose and denote conjugation by and respectively. Let denote the automorphism that sends to and to . Then, is the inner automorphism group and is the automorphism group.
The automorphism fixes and while interchanging and . The automorphism interchanges and while also interchanging and . The automorphism fixes and while interchanging and . The automorphism interchanges and and also interchanges and .
|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 and ) that are normal in the whole group.|
|subnormal subgroup||Yes||follows from being 2-subnormal, also from being subgroup of nilpotent group.|