A4 in 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) alternating group:A4 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:

$\!H=H_{5}=\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,3),(1,3,2),(1,2,4),(1,4,2),(1,3,4),(1,4,3),(2,3,4),(2,4,3)\}$

$H$ is the alternating group on the set $\{1,2,3,4\}$ fixing the point 5.

$H$ has five conjugate subgroups (including the subgroup itself) in $G$, based on the choice of fixed point:

Fixed point	Subgroup name here	Subgroup
1	$H_{1}$	$\!\{(),(2,3)(4,5),(2,4)(3,5),(2,5)(3,4),(2,3,4),(2,4,3),(2,3,5),(2,5,3),(2,4,5),(2,5,4),(3,4,5),(3,5,4)\}$
2	$H_{2}$	$\!\{(),(1,3)(4,5),(1,4)(3,5),(1,5)(3,4),(1,3,4),(1,4,3),(1,3,5),(1,5,3),(1,4,5),(1,5,4),(3,4,5),(3,5,4)\}$
3	$H_{3}$	$\!\{(),(1,2)(4,5),(1,4)(2,5),(1,5)(2,4),(1,2,4),(1,4,2),(1,2,5),(1,5,2),(1,4,5),(1,5,4),(2,4,5),(2,5,4)\}$
4	$H_{4}$	$\!\{(),(1,2)(3,5),(1,3)(2,5),(1,5)(2,3),(1,2,3),(1,3,2),(1,2,5),(1,5,2),(1,3,5),(1,5,3),(2,3,5),(2,5,3)\}$
5	$H_{5}$	$\!\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,3),(1,3,2),(1,2,4),(1,4,2),(1,3,4),(1,4,3),(2,3,4),(2,4,3)\}$

Arithmetic functions

Function	Value	Explanation
order of the whole group	60	$5!/2=120/2=60$. See alternating group:A5.
order of the subgroup	12
index of the subgroup	5
size of conjugacy class of subgroup	5
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	the subgroup itself	current page	alternating group:A4
centralizer	trivial subgroup	--	trivial 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 normalizer 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, hence all normalizers of $p$-Sylow subgroups are also conjugate.	The 2-Sylow subgroup is V4 in A5, and its normalizer is precisely this.

Related subgroups

Intermediate subgroups

There are no intermediate subgroups, since the subgroup is a maximal subgroup.

Smaller subgroups

Value of smaller subgroup (descriptive)	Isomorphism class of smaller subgroup	Smaller subgroup in subgroup	Smaller subgroup in whole group
$\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$	Klein-four group	V4 in A4	V4 in A5
$\{(),(1,2,3),(1,3,2)\}$, $\{(),(1,2,4),(1,4,2)\}$, $\{(),(1,3,4),(1,4,3)\}$, $\{(),(2,3,4),(2,4,3)\}$	cyclic group:Z3	A3 in A4	A3 in A5
$\{(),(1,2)(3,4)\}$, $\{(),(1,3)(2,4)\}$, $\{(),(1,4)(2,3)\}$	cyclic group:Z2	subgroup generated by double transposition in A4	subgroup generated by double transposition in A5

Description in alternative interpretations of the whole group

Description of $G$	Corresponding description of $H$
special linear group of degree two over field:F4, i.e., $SL(2,4)$	Borel subgroup, i.e., subgroup of upper-triangular invertible matrices where the two diagonal entries are mutual inverses. See more at Borel subgroup of special linear group of degree two.
projective special linear group of degree two over field:F5, i.e., $PSL(2,5)$	Unclear/nothing concise (?)

Subgroup properties

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
abnormal subgroup	For any element $g\in G$, we have $g\in \langle H,gHg^{-1}\rangle$	Yes	Follows from being a Sylow normalizer, see Sylow normalizer implies abnormal
weakly abnormal subgroup		Yes	Follows from being abnormal
self-normalizing subgroup	equals normalizer in the whole group	Yes
self-centralizing subgroup	contains its centralizer in the whole group	Yes
subgroup whose join with any distinct conjugate is the whole group	join of the subgroup with any distinct conjugate subgroup is the whole group	Yes
maximal subgroup	no proper subgroup containing it	Yes		Note that in a finite solvable group, any maximal subgroup has prime power index (see maximal subgroup has prime power index in finite solvable group). The fact that we have a maximal subgroup here whose index is not a prime power is consistent with the fact that alternating group:A5 is not solvable.
pronormal subgroup	any conjugate to it is conjugate in their join	Yes
weakly pronormal subgroup		Yes
paranormal subgroup		Yes
polynormal subgroup		Yes

Other properties

Property	Meaning	Satisfied?	Explanation	Comment
Hall subgroup	order and index are relatively prime	Yes	$\{2,3\}$-Hall subgroup. Also, order (12) and index (5) are relatively prime.
p-complement	complement of a $p$-Sylow subgroup	Yes	$p$-complement for $p=5$