D8 in S4
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) dihedral group:D8 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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This article is about the subgroup in the group
, where
is symmetric group:S4, i.e., the symmetric group on the set
, and
is the subgroup:
is a 2-Sylow subgroup of
and is isomorphic to dihedral group:D8. It has two other conjugate subgroups, which are given below:
and:
Contents
Complements
The permutable complements to (and also to each of its conjugates) are precisely the subgroups of order three, namely A3 in S4 and its conjugates. Each of these is a conjugate to each of the conjugates of
:
In addition, any S2 in S4 is a lattice complement but not a permutable complement to two of the three conjugates of , namely the ones that do not contain it.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
permutably complemented subgroup | has a permutable complement | Yes | A3 in S4 | |
lattice-complemented subgroup | has a lattice complement | Yes | A3 in S4, or S2 in S4 | |
retract | has a normal complement | No | ||
complemented normal subgroup | No | not a normal subgroup |
Arithmetic functions
Function | Value | Explanation | Comment |
---|---|---|---|
order of the group | 24 | ||
order of the subgroup | 8 | ||
index of the subgroup | 3 | ||
size of conjugacy class | 3 | ||
number of conjugacy classes in automorphism class | 1 |
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 | the subgroup itself | (current page) | dihedral group:D8 |
centralizer | ![]() |
subgroup generated by double transposition in S4 | cyclic group:Z2 |
normal core | ![]() |
normal Klein four-subgroup of symmetric group:S4 | Klein four-group |
normal closure | the whole group | -- | symmetric group:S4 |
characteristic core | ![]() |
normal Klein four-subgroup of symmetric group:S4 | Klein four-group |
characteristic closure | the whole group | -- | symmetric group:S4 |
Subgroup properties
Resemblance-based properties
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
order-conjugate subgroup | conjugate to any subgroup of the same order | Yes | Sylow implies order-conjugate | |
order-dominating subgroup | any subgroup of the whole group whose order divides the order of ![]() ![]() |
Yes | Sylow implies order-dominating | |
order-dominated subgroup | any subgroup of the whole group whose order is a multiple of the order of ![]() ![]() |
Yes | Sylow implies order-dominated | |
order-isomorphic subgroup | isomorphic to any subgroup of the group of the same order | Yes | (via order-conjugate) | |
isomorph-automorphic subgroup | Yes | (via order-conjugate) | ||
automorph-conjugate subgroup | Yes | (via order-conjugate) | ||
order-automorphic subgroup | Yes | (via order-conjugate) | ||
isomorph-conjugate subgroup | Yes | (via order-conjugate) |
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | equals all its conjugate subgroups | No | (see other conjugate subgroups) | |
subnormal subgroup | No | |||
self-normalizing subgroup | equals its normalizer in the whole group | Yes | ||
abnormal subgroup | Yes | |||
weakly abnormal subgroup | Yes | |||
contranormal subgroup | Yes | |||
maximal subgroup | Yes | |||
pronormal subgroup | Yes | Sylow implies pronormal | ||
weakly pronormal subgroup | Yes | (via pronormal) |
Fusion system on subgroup
The subgroup embedding induces the non-inner non-simple fusion system for dihedral group:D8.
GAP implementation
Construction of subgroup given group as a black box
Suppose we are already given a group that we know to be isomorphic to symmetric group:S4. Then, the subgroup
can be constructed using SylowSubgroup as follows:
H := SylowSubgroup(G,2);
Construction of group-subgroup pair
The group and subgroup pair can be defined using GAP's SymmetricGroup and SylowSubgroup functions as follows:
G := SymmetricGroup(4); H := SylowSubgroup(G,2);
Note that this doesn't output the subgroup used on this page, but rather the conjugate subgroup
. However all relevant properties are invariant under conjugation, so this does not matter.