A3 in S5

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) cyclic group:Z3 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Let $G$ be the alternating group:A5, i.e., the alternating group (the group of all permutations) on the set $\{1,2,3,4,5\}$. $G$ has order $5!=120$.

Consider the subgroup:

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

$H$ has a total of 10 conjugate subgroups (including $H$ itself) and the subgroups are parametrized by subsets of size 3 in $\{1,2,3,4,5\}$ describing the support of the 3-cycles. The complementary subset of size two is fixed point-wise by that conjugate subgroup:

Subset of size 3	Complementary subset of size 2 fixed point-wise by the subgroup	Conjugate of $H$
1,2,3	4,5	$\!\{(),(1,2,3),(1,3,2)\}$
1,2,4	3,5	$\!\{(),(1,2,4),(1,4,2)\}$
1,2,5	3,4	$\!\{(),(1,2,5),(1,5,2)\}$
1,3,4	2,5	$\!\{(),(1,3,4),(1,4,3)\}$
1,3,5	2,4	$\!\{(),(1,3,5),(1,5,3)\}$
1,4,5	2,3	$\!\{(),(1,4,5),(1,5,4)\}$
2,3,4	1,5	$\!\{(),(2,3,4),(2,4,3)\}$
2,3,5	1,4	$\!\{(),(2,3,5),(2,5,3)\}$
2,4,5	1,3	$\!\{(),(2,4,5),(2,5,4)\}$
3,4,5	1,2	$\!\{(),(3,4,5),(3,5,4)\}$

Each of these subgroups is isomorphic to cyclic group:Z3.

Arithmetic functions

Function	Value	Explanation
order of the whole group	120	$5!=120$. See symmetric group:S5.
[[order of a group|order of the subgroup	3
index of the subgroup	40	Follows from Lagrange's theorem
size of conjugacy class of subgroups = index of normalizer	10
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	$\langle (1,2,3),(1,2),(4,5)\rangle$	direct product of S3 and S2 in S5	dihedral group:D12
centralizer	$\langle (1,2,3),(4,5)\rangle$	Z6 in S5	cyclic group:Z6
normal core	trivial subgroup	--	trivial group
normal closure	the alternating subgroup	A5 in S5	alternating group:A5
characteristic core	trivial subgroup	--	trivial group
characteristic closure	the alternating subgroup	A5 in S5	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 $p=3$	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 3, which is the largest power of 3 dividing the order of the group.