# 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 $p = 5$