Klein four-subgroup of alternating group:A5

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) alternating group:A5 (see subgroup structure of alternating group:A5).
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Let $G$ be the alternating group:A5, i.e., the alternating group (the group of even permutations) on the set $\{ 1,2,3,4,5 \}$. $G$ has order $5!/2 = 60$.

Consider the subgroup:

$H = H_5 := \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$

This subgroup is isomorphic to the Klein four-group. There are five conjugates (including the subgroup itself) depending on which of the five points $1,2,3,4,5$ is fixed:

Fixed point	Subgroup name here	Elements of subgroup
1	$H_1$	$\{ (), (2,3)(4,5), (2,4)(3,5), (2,5)(3,4) \}$
2	$H_2$	$\{ (), (1,3)(4,5), (1,4)(3,5), (1,5)(3,4) \}$
3	$H_3$	$\{ (), (1,2)(4,5), (1,4)(2,5), (1,5)(2,4) \}$
4	$H_4$	$\{ (), (1,2)(3,5), (1,3)(2,5), (1,5)(2,3) \}$
5	$H_5$	$\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$

Arithmetic functions

Function	Value	Explanation
order of the whole group	60	The order is $5!/2 = 120/2 =60$. See alternating group:A5.
order of the subgroup	4
index of the subgroup	15
size of conjugacy class of subgroups (equal to index of normalizer)	3
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

In the table below, we provide values specific to $H$.

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	$\langle (1,2)(3,4), (1,2,3) \rangle$	A4 in A5	alternating group:A4
centralizer	the subgroup itself	current page	Klein four-group
normal core	trivial subgroup	--	trivial group
normal closure	the whole group	--	alternating group:A5
characteristic core	trivial subgroup	--	trivial group
characteristic closure	the whole group	--	alternating group:A5

Conjugacy class-defining functions

Conjugacy class-defining function	What it means in general	Why it takes this value
Sylow subgroup for the prime $p = 2$	A $p$-Sylow subgroup is a subgroup whose order is a power of $p$ and index is relatively prime to $p$. Sylow subgroups exist and Sylow implies order-conjugate, i.e., all $p$-Sylow subgroups are conjugate to each other.	The order of this subgroup is 4, which is the largest power of 2 dividing the order of the group.

Related subgroups

Intermediate subgroups

Value of intermediate subgroup (descriptive)	Isomorphism class of intermediate subgroup	Number of conjugacy classes of intermediate subgroup fixing subgroup and whole group	Subgroup in intermediate subgroup	Intermediate subgroup in whole group
$\langle (1,2)(3,4), (1,2,3) \rangle$	alternating group:A4	1	V4 in A4	A4 in A5

Smaller subgroups

Value of smaller subgroup (descriptive)	Isomorphism class of smaller subgroup	Number of conjugacy classes of smaller subgroup fixing subgroup and whole group	Smaller subgroup in subgroup	Smaller subgroup in whole group
$\{ (), (1,2)(3,4) \}, \{ (), (1,3)(2,4) \}, \{ (), (1,4)(2,3) \}$	cyclic group:Z2	1	Z2 in V4	subgroup generated by double transposition in A5

Subgroup properties

Sylow and corollaries

The subgroup is a 2-Sylow subgroup, so many properties follow as a corollary of that.

Property	Meaning	Why being Sylow implies the property
order-conjugate subgroup	conjugate to any subgroup of the same order	Sylow implies order-conjugate
isomorph-conjugate subgroup	conjugate to any subgroup isomorphic to it	(via order-conjugate)
automorph-conjugate subgroup	conjugate to any subgroup automorphic to it	(via isomorph-conjugate)
intermediately isomorph-conjugate subgroup	conjugate to any subgroup isomorphic to it inside any intermediate subgroup	Sylow implies intermediately isomorph-conjugate
intermediately automorph-conjugate subgroup	automorph-conjugate in every intermediate subgroup	(via intermediately isomorph-conjugate)
pronormal subgroup	conjugate to any conjugate subgroup in their join	Sylow implies pronormal
weakly pronormal subgroup		(via pronormal)
paranormal subgroup		(via pronormal)
polynormal subgroup		(via pronormal)

Normality-related properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No	(see above for other conjugate subgroups)
2-subnormal subgroup	normal subgroup in its normal closure	No
subnormal subgroup	series from subgroup to whole group, each normal in next	No
contranormal subgroup	normal closure is whole group	Yes
self-normalizing subgroup	equals its normalizer in the whole group	No	normalizer is A4 in A5
self-centralizing subgroup	contains its centralizer in the whole group	Yes