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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Contents
Definition
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.
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 .
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 | ![]() ![]() |
-- | dihedral group:D8 |
Related subgroups
Intermediate subgroups
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 |
Subgroup properties
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
.
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 ![]() ![]() |
|
subnormal subgroup | Yes | follows from being 2-subnormal, also from being subgroup of nilpotent group. |