Klein four-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) Klein four-group and the group is (up to isomorphism) dihedral group:D8 (see subgroup structure of dihedral group:D8).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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This article discusses the dihedral group of order eight (see details on the subgroup structure) and the two Klein four-subgroups of this group. We call the dihedral group , and use the following presentation:
and we set:
.
As full lists:
Contents
Cosets
Both subgroups have index two and are hence normal subgroups. Each of them has two cosets: the subgroup itself and the rest of the group.
The cosets of are:
The cosets of are:
Complements
COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)
has two permutable complements:
and
, both of which are conjugate subgroups and subgroups of
.
has two permutable complements:
and
, both of which are conjugate subgroups and are subgroups of
.
The four subgroups of order two arising as complements in this way are all automorphic subgroups. For information on these, see non-normal subgroups of dihedral group:D8.
For convenience, we refer to below, though
behaves the same way.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
complemented normal subgroup | normal subgroup with permutable complement | Yes | see above | |
permutably complemented subgroup | subgroup with permutable complement | Yes | (via complemented normal) | |
lattice-complemented subgroup | subgroup with lattice complement | Yes | (via permutably complemented) | |
retract | has a normal complement | No | ||
direct factor | normal subgroup with normal complement | No |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 8 | |
order of subgroup | 4 | |
index | 2 | |
size of conjugacy class | 1 | |
number of conjugacy classes in automorphism class | 2 |
Effect of subgroup operators
In the table below, we provide values specific to .
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | whole group ![]() |
dihedral group:D8 | |
centralizer | the subgroup itself | current page | Klein four-group |
normal core | the subgroup itself | current page | Klein four-group |
normal closure | the subgroup itself | current page | Klein four-group |
characteristic core | ![]() |
center of dihedral group:D8 | cyclic group:Z2 |
characteristic closure | whole group | dihedral group:D8 | |
commutator with whole group | ![]() |
center of dihedral group:D8 | cyclic group:Z2 |
Automorphism class-defining functions
Below are some functions that start with the whole group as a black box group and give the unique automorphism class comprising the two Klein four-subgroups. Note that these functions are not guaranteed to always give a single automorphism class of subgroups for a general group.
Function | Where does it make sense in general? | Always gives unique automorphism class? | Why it gives these subgroups? |
---|---|---|---|
elementary abelian subgroup of maximum order | typically, context of group of prime power order | No | By inspection of subgroup structure |
abelian subgroup of maximum rank | typically, context of group of prime power order | No | By inspection of subgroup structure |
non-characteristic normal subgroup | any group | No | By inspection of subgroup structure |
Related subgroups
Intermediate subgroups
There are no properly intermediate subgroups because the subgroup is a maximal subgroup.
Smaller subgroups
Below we discuss subgroups inside .
Value of smaller subgroup (descriptive) | Isomorphism class of smaller subgroup | Smaller subgroup in subgroup | Smaller subgroup in whole group |
---|---|---|---|
![]() |
cyclic group:Z2 | Z2 in V4 | center of dihedral group:D8 |
![]() |
cyclic group:Z2 | Z2 in V4 | non-normal subgroups of dihedral group:D8 |
![]() |
cyclic group:Z2 | Z2 in V4 | non-normal subgroups of dihedral group:D8 |
Images under quotient maps
The discussion below is for .
Kernel of quotient map (descriptive) | Kernel of quotient map (link) | Image of whole group as group | Image of subgroup as group | Embedding (link) |
---|---|---|---|---|
![]() |
cyclic maximal subgroup of dihedral group:D8 | cyclic group:Z2 | cyclic group:Z2 | Z2 in itself |
![]() |
Klein four-subgroups of dihedral group:D8 | cyclic group:Z2 | cyclic group:Z2 | Z2 in itself |
![]() |
center of dihedral group:D8 | Klein four-group | cyclic group:Z2 | Z2 in V4 |
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. It turns out that
and
, while
and
preserve both
and
. Note that the automorphism group is isomorphic to dihedral group:D8 and the inner automorphism group to the Klein four-group.
Note that since sends
to
, they both satisfy the same subgroup properties. For convenience, we use notation and symbols for
.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | invariant under inner automorphisms | Yes | subgroup of index two is normal | |
coprime automorphism-invariant subgroup | invariant under automorphisms of coprime order to group | Yes | there are no nontrivial automorphisms of coprime order | |
coprime automorphism-invariant normal subgroup | normal and coprime automorphism-invariant | Yes | ||
characteristic subgroup | invariant under all automorphisms | No | ![]() |
|
cofactorial automorphism-invariant subgroup | invariant under all automorphisms whose order has prime factors only among those of the group | No | ![]() ![]() |
|
fully invariant subgroup | invariant under all endomorphisms | No | (via characteristic) |
Advanced properties based on invariance/resemblance
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
order-normal subgroup | every subgroup of the same order is normal | Yes | maximal subgroup of group of prime power order is normal | |
isomorph-normal subgroup | every isomorphic subgroup is normal | Yes | ||
isomorph-automorphic subgroup | every isomorphic subgroup is automorphic | Yes | ||
order-isomorphic subgroup | isomorphic to all subgroups of the same order | No | Cyclic subgroup ![]() |
Property | Meaning | Satisfied? | Explanation |
---|---|---|---|
central factor | product with centralizer is whole group | No | The subgroup is a proper self-centralizing subgroup. |
conjugacy-closed subgroup | any two elements of the subgroup conjugate in the whole group are conjugate in the subgroup | No | The elements ![]() ![]() |
transitively normal subgroup | every normal subgroup of the subgroup is normal in the whole group | No | The subgroup ![]() ![]() ![]() ![]() |
Generic maximality notions
and
are both maximal subgroups of the group of prime power order
. Thus, they satisfy all these properties: maximal normal subgroup, maximal subgroup, subgroup of index two, order-normal subgroup, isomorph-normal subgroup, maximal subgroup of finite nilpotent group.
As already discussed, they are both also coprime automorphism-invariant, hence they are isomorph-normal coprime automorphism-invariant subgroup of group of prime power order. In particular, they are both fusion system-relatively weakly closed subgroups and thus Sylow-relatively weakly closed subgroups.
Abelian subgroups of maximum order
and
are both abelian subgroups of maximum order. There is one more abelian subgroup of maximum order, namely the cyclic subgroup
.
Elementary abelian subgroups of maximum order
and
are both elementary abelian subgroups of maximum order in
. There are no others.
Abelian subgroups of maximum rank
and
are both abelian subgroups of maximum rank in
. There are no others.
GAP implementation
Finding these subgroups inside a black-box dihedral group
Suppose is a group that we know is isomorphic to dihedral group:D8. We can create a two-element list of the Klein four-subgroups of
using the command:
H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [4,2]);
We can then set:
H1 := H[1]; H2 := H[2];
Constructing the dihedral group and the two subgroups
This can be achieved by the sequence of commands:
G := DihedralGroup(8); H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [4,2]); H1 := H[1]; H2 := H[2];