A5 in S5
From Groupprops
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:A5 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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Definition
We define as symmetric group:S5 -- for concreteness, the symmetric group on the set .
is alternating group:A5 -- the subgroup of comprising even permutations. A criterion for a permutation to be even, based on cycle decomposition, is that the number of cycles of even length should be even.
The quotient group is cyclic group:Z2.
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of group | 120 | Order is |
order of subgroup | 60 | Order is |
index of subgroup | 2 | For degree , the alternating group of degree has index two in the symmetric group of degree , since it has two cosets: itself (the even permutations) and the odd permutations. |
Subgroup-defining functions
Subgroup-defining function | Meaning in general | Why it takes this value | GAP verification (set G := SymmetricGroup(5); H := AlternatingGroup(5);) |
---|---|---|---|
derived subgroup (also called commutator subgroup) | subgroup generated by commutators | Since the quotient is abelian, it contains the derived subgroup. Further, is simple non-abelian, so the derived subgroup cannot be smaller. | H = DerivedSubgroup(G); using DerivedSubgroup |
hypocenter | stable member of lower central series (transfinite if necessary) | is the derived subgroup, and | |
perfect core | stable member of derived series (transfinite if necessary); alternatively, join of all perfect subgroups | is perfect, and the only bigger subgroup is , which is not perfect because | |
Jacobson radical | intersection of all maximal normal subgroups | is the unique maximal normal subgroup ( is a one-headed group). This can also be seen from the fact that it is simple, so no other maximal normal subgroup can intersect it nontrivially, and it has no normal complement, so no maximal normal subgroup can intersect it nontrivially. | |
socle | join of all minimal normal subgroups | It is the unique minimal normal subgroup ( is a monolithic group). | H = Socle(G); using Socle |
3-Sylow closure | normal closure of any 3-Sylow subgroup | Any 3-Sylow subgroup lies within . Since is a minimal normal subgroup, it must be the normal closure of that subgroup. | |
5-Sylow closure | normal closure of any 5-Sylow subgroup | Any 5-Sylow subgroup lies within . Since is a minimal normal subgroup, it must be the normal closure of that subgroup. |
GAP implementation
We can define the group and subgroup naturally, using the SymmetricGroup and AlternatingGroup functions:
G := SymmetricGroup(5); H := AlternatingGroup(5);