A3 in S5

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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.