Klein four-subgroup of alternating group:A5

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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) 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