S4 in S5: Difference between revisions

Latest revision as of 16:36, 26 August 2024

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) symmetric group:S4 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Definition

The group $G$ is taken as symmetric group:S5: the symmetric group of degree five. For concreteness, we take $G$ as the symmetric group on the set ${1, 2, 3, 4, 5}$ .

We take $H = H_{5}$ as the subgroup fixing ${5}$ , so $H$ is symmetric group:S4 acting on the set ${1, 2, 3, 4}$ .

$H$ has four other conjugate subgroups, each corresponding to a different fixed point:

$H_{1}$ is the subgroup fixing ${1}$ , and is the symmetric group on the set ${2, 3, 4, 5}$ .
$H_{2}$ is the subgroup fixing ${2}$ , and is the symmetric group on the set ${1, 3, 4, 5}$ .
$H_{3}$ is the subgroup fixing ${3}$ , and is the symmetric group on the set ${1, 2, 4, 5}$ .
$H_{4}$ is the subgroup fixing ${4}$ , and is the symmetric group on the set ${1, 2, 3, 5}$ .
$H = H_{5}$ is the subgroup fixing ${5}$ , and is the symmetric group on the set ${1, 2, 3, 4}$ .

Arithmetic functions

Function	Value	Explanation
order of group	120
order of subgroup	24
index of subgroup	5
size of conjugacy class of subgroup	5	conjugacy class is made of those listed above
number of conjugacy classes in automorphism class of subgroup	1
size of automorphism class of subgroup	5

Subgroup properties

Other properties

Property	Meaning	Satisfied?	Explanation
Hall subgroup	order and index are relatively prime	Yes	${2, 3}$ -Hall subgroup. Also, order (24) and index (5) are relatively prime.
normal subgroup		No
p-complement	complement of a $p$ -Sylow subgroup	Yes	$p$ -complement for $p = 5$

GAP implementation

The group-subgroup pair can be constructed as follows:

G := SymmetricGroup(5); H := SymmetricGroup(4);

@@ Line 28: / Line 28: @@
 | [[index of a subgroup|index]] of subgroup || [[arithmetic function value::index of a subgroup;5|5]] ||
 |-
-| [[size of conjugacy class of subgroup]] || [[arithmetic function value::size of conjugacy class of subgroup;5|5]] ||
+| [[size of conjugacy class of subgroup]] || [[arithmetic function value::size of conjugacy class of subgroup;5|5]] || conjugacy class is made of those listed above
 |-
-| [[number of conjugacy classes in automorphism class of subgroup]] || [[arithmetic function value::number of conjugacy classes in automorphism clsas of subgroup;1|1]]||
+| [[number of conjugacy classes in automorphism class of subgroup]] || [[arithmetic function value::number of conjugacy classes in automorphism class of subgroup;1|1]]||
 |-
 | [[size of automorphism class of subgroup]] || [[arithmetic function value::size of automorphism class of subgroup;5|5]] ||
@@ Line 43: / Line 43: @@
 |-
 | [[satisfies property::Hall subgroup]] || order and index are relatively prime || Yes || <math>\{ 2,3 \}</math>-Hall subgroup. Also, order (24) and index (5) are relatively prime. ||
+|-
+| [[satisfies property::normal subgroup]] ||  || No ||  ||
 |-
 | [[satisfies property::p-complement]] || [[permutable complements|complement]] of a <math>p</math>-[[Sylow subgroup]] || Yes || <math>p</math>-complement for <math>p = 5</math> ||
 |}
 ==GAP implementation==
@@ Line 52: / Line 55: @@
 <tt>G := SymmetricGroup(5); H := SymmetricGroup(4);</tt>
+==See also==
+* [[S3 in S5]]