Difference between revisions of "A5 in S5"
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| order of subgroup || 60 || Order is <math>5!/2 = 120/2 = 60</math> | | order of subgroup || 60 || Order is <math>5!/2 = 120/2 = 60</math> | ||
|- | |- | ||
− | | [[index of a subgroup|index]] of subgroup || 2 || For <math>n \ge 2</math>, the alternating group has index two in the symmetric group, since it has two cosets: itself (the even permutations) and the odd permutations. | + | | [[index of a subgroup|index]] of subgroup || 2 || For degree <math>n \ge 2</math>, the alternating group of degree <math>n</math> has index two in the symmetric group of degree <math>n</math>, since it has two cosets: itself (the even permutations) and the odd permutations. |
|} | |} | ||
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! [[Subgroup-defining function]] !! Meaning in general !! Why it takes this value !! GAP verification (set <tt>G := SymmetricGroup(5); H := AlternatingGroup(5);</tt>) | ! [[Subgroup-defining function]] !! Meaning in general !! Why it takes this value !! GAP verification (set <tt>G := SymmetricGroup(5); H := AlternatingGroup(5);</tt>) | ||
|- | |- | ||
− | | [[arises as subgroup-defining function::derived subgroup]] (also called commutator subgroup) || subgroup generated by | + | | [[arises as subgroup-defining function::derived subgroup]] (also called commutator subgroup) || subgroup generated by [[commutator]]s || Since the quotient is abelian, it contains the derived subgroup. Further, <math>H</math> is simple non-abelian, so the derived subgroup cannot be smaller. || {{GAP verify sdf|DerivedSubgroup}} |
|- | |- | ||
| [[arises as subgroup-defining function::hypocenter]] || stable member of [[lower central series]] (transfinite if necessary) || <math>H</math> is the derived subgroup, and <math>[G,H] = H</math> || | | [[arises as subgroup-defining function::hypocenter]] || stable member of [[lower central series]] (transfinite if necessary) || <math>H</math> is the derived subgroup, and <math>[G,H] = H</math> || | ||
|- | |- | ||
| [[arises as subgroup-defining function::perfect core]] || stable member of [[derived series]] (transfinite if necessary); alternatively, join of all perfect subgroups || <math>H</math> is perfect, and the only bigger subgroup is <math>G</math>, which is not perfect because <math>[G,G] = H</math> || | | [[arises as subgroup-defining function::perfect core]] || stable member of [[derived series]] (transfinite if necessary); alternatively, join of all perfect subgroups || <math>H</math> is perfect, and the only bigger subgroup is <math>G</math>, which is not perfect because <math>[G,G] = H</math> || | ||
+ | |- | ||
+ | | [[arises as subgroup-defining function::Jacobson radical]] || intersection of all [[maximal normal subgroup]]s || <math>H</math> is the unique [[maximal normal subgroup]] (<matH>G</math> 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. || | ||
+ | |- | ||
+ | | [[arises as subgroup-defining function::socle]] || join of all [[minimal normal subgroup]]s || It is the unique [[minimal normal subgroup]] (<math>G</math> is a [[monolithic group]]). || {{GAP verify sdf|Socle}} | ||
+ | |- | ||
+ | | 3-[[arises as subgroup-defining function::Sylow closure]] || [[normal closure]] of any 3-Sylow subgroup || Any 3-Sylow subgroup lies within <math>H</math>. Since <math>H</math> is a [[minimal normal subgroup]], it must be the normal closure of that subgroup. || | ||
+ | |- | ||
+ | | 5-[[arises as subgroup-defining function::Sylow closure]] || [[normal closure]] of any 5-Sylow subgroup || Any 5-Sylow subgroup lies within <math>H</math>. Since <math>H</math> is a [[minimal normal subgroup]], it must be the normal closure of that subgroup. || | ||
+ | |} | ||
+ | |||
+ | ==Description in alternative interpretations of the whole group== | ||
+ | |||
+ | {| class="sortable" border="1" | ||
+ | ! Interpretation of <math>G</math> !! Corresponding interpretation of <math>H</math> | ||
+ | |- | ||
+ | | Symmetric group of degree five. || Alternating group of degree five. | ||
+ | |- | ||
+ | | <math>PGL(2,5)</math>, the [[projective general linear group of degree two]] over [[field:F5]] || <math>PSL(2,5)</math>, the [[projective special linear group of degree two]] over [[field:F5]]. In other words, the cosets mod the center of the linear transformations of determinant 1. | ||
+ | |- | ||
+ | | <math>P\Gamma L(2,4)</math>, the [[projective semilinear group of degree two]] over [[field:F4]]. || <math>PGL(2,4)</math> (same as <math>PSL(2,4)</math>). In other words, the cosets mod scalars of the transformations that are actually ''linear'', not merely semilinear. | ||
|} | |} | ||
+ | |||
==GAP implementation== | ==GAP implementation== | ||
− | + | ===Finding this subgroup inside the group as a black box=== | |
+ | |||
+ | Here, a group <math>G</math> that we know to be isomorphic to the symmetric group of degree five is given, and we need to locate in that the alternating group of degree four. Different ways of constructing/locating this subgroup are given below. | ||
+ | |||
+ | {| class="sortable" border="1" | ||
+ | ! Description !! Functions used | ||
+ | |- | ||
+ | | <tt>DerivedSubgroup(G)</tt> || [[GAP:DerivedSubgroup|DerivedSubgroup]] | ||
+ | |- | ||
+ | | <tt>Socle(G)</tt> || [[GAP:Socle|Socle]] | ||
+ | |} | ||
+ | |||
+ | To assign <math>H</math> to any of these, do <tt>H :=</tt> followed by that. For instance: | ||
+ | |||
+ | <tt>H := DerivedSubgroup(G);</tt> | ||
+ | |||
+ | ===Constructing the group and the subgroup=== | ||
+ | |||
+ | Because of GAP's native implementation of symmetric groups, this can be easily achieved using [[GAP:SymmetricGroup|SymmetricGroup]] and [[GAP:AlternatingGroup|AlternatingGroup]]: | ||
+ | |||
+ | <pre>gap> G := SymmetricGroup(5);; | ||
+ | gap> H := AlternatingGroup(5);;</pre> | ||
− | + | Note that double semicolons have been used to suppress confirmatory output, but you may prefer to use single semicolons. |
Latest revision as of 19:08, 30 April 2012
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.
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups
Contents
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. |
Description in alternative interpretations of the whole group
Interpretation of | Corresponding interpretation of |
---|---|
Symmetric group of degree five. | Alternating group of degree five. |
, the projective general linear group of degree two over field:F5 | , the projective special linear group of degree two over field:F5. In other words, the cosets mod the center of the linear transformations of determinant 1. |
, the projective semilinear group of degree two over field:F4. | (same as ). In other words, the cosets mod scalars of the transformations that are actually linear, not merely semilinear. |
GAP implementation
Finding this subgroup inside the group as a black box
Here, a group that we know to be isomorphic to the symmetric group of degree five is given, and we need to locate in that the alternating group of degree four. Different ways of constructing/locating this subgroup are given below.
Description | Functions used |
---|---|
DerivedSubgroup(G) | DerivedSubgroup |
Socle(G) | Socle |
To assign to any of these, do H := followed by that. For instance:
H := DerivedSubgroup(G);
Constructing the group and the subgroup
Because of GAP's native implementation of symmetric groups, this can be easily achieved using SymmetricGroup and AlternatingGroup:
gap> G := SymmetricGroup(5);; gap> H := AlternatingGroup(5);;
Note that double semicolons have been used to suppress confirmatory output, but you may prefer to use single semicolons.