A3 in S3
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:S3 (see subgroup structure of symmetric group:S3).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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We consider the subgroup in the group defined as follows.
is the symmetric group of degree three, which, for concreteness, we take as the symmetric group on the set .
is the subgroup of comprising the identity element and the two 3-cycles. It is thus the subgroup of all even permutations, i.e., the alternating group . Explicitly:
is a normal subgroup and in fact a characteristic subgroup of . It is the unique -Sylow subgroup of .
See also subgroup structure of symmetric group:S3.
Contents
Cosets
The subgroup is a subgroup of index two, and hence has two cosets -- the subgroup itself and the complement of the subgroup in the group:
In particular, the subgroup is a normal subgroup -- every left coset is a right coset and vice versa. (see also index two implies normal).
Complements
COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:
TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)
FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)
is a complemented normal subgroup of . There are three possibilities for its permutable complement, all of which are conjugate subgroups. This can also be seen by the Schur-Zassenhaus theorem since is a normal Sylow subgroup (i.e., normal and a Sylow subgroup) and hence a normal Hall subgroup (i.e., normal and a Hall subgroup).
The three complements are:
For information on these as subgroups, see S2 in S3.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
complemented normal subgroup | normal subgroup having a permutable complement | Yes | ||
complemented characteristic subgroup | characteristic subgroup having a permutable complement | Yes | ||
permutably complemented subgroup | has a permutable complement | Yes | ||
lattice-complemented subgroup | has a lattice complement | Yes | ||
retract | has a normal complement | No | ||
direct factor | normal subgroup having a normal complement | No |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 6 | |
order of subgroup | 3 | |
index | 2 | |
size of conjugacy class | 1 | |
number of conjugacy classes in automorphism class | 1 |
Effect of subgroup operators
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | the whole group | -- | symmetric group:S3 |
centralizer | the subgroup itself | current page | cyclic group:Z3 |
normal core | the subgroup itself | current page | cyclic group:Z3 |
normal closure | the subgroup itself | current page | cyclic group:Z3 |
characteristic core | the subgroup itself | current page | cyclic group:Z3 |
characteristic closure | the subgroup itself | current page | cyclic group:Z3 |
commutator with whole group | the subgroup itself | current page | cyclic group:Z3 |
Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as a result of many common subgroup-defining functions on the whole group. Some of these are given below.
Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|
derived subgroup | subgroup generated by commutators of all pairs of group elements, smallest subgroup with abelian quotient | The quotient is cyclic group:Z2, which is abelian; no other subgroup has abelian quotient. We can also explicitly compute all commutators -- these are precisely the identity element and the two 3-cycles. |
hypocenter | subgroup at which the lower central series of the whole group stabilizes | It is the derived subgroup and its commutator with the whole group equals itself. |
socle | join of all minimal normal subgroups | The subgroup is the unique minimal normal subgroup (i.e., monolith) -- the group is a monolithic group |
Fitting subgroup | join of all nilpotent normal subgroups | The subgroup is the unique nontrivial abelian normal subgroup |
3-Sylow core | largest normal subgroup whose order is a power of 3; normal core of any 3-Sylow subgroup | The subgroup is the unique normal 3-Sylow subgroup |
3-Sylow closure | normal closure of any 3-Sylow subgroup | The subgroup is the unique normal 3-Sylow subgroup |
Brauer core | largest normal subgroup of odd order | (same reason as 3-Sylow core; order has two prime factors 2 and 3) |
Jacobson radical | intersection of all maximal normal subgroups | The subgroup is the unique maximal normal subgroup -- the group is a one-headed group |
Description in alternative interpretations of the whole group
Interpretation of | Corresponding interpretation of |
---|---|
As the dihedral group of degree three, order six | The cyclic part comprising rotations, i.e., orientation-preserving elements. |
As the general linear group of degree two over field:F2 | The subgroup comprising the semisimple elements, although the ones of order three are not diagonalizable over field:F2 and can be diagonalized only over field:F4. Note that it is a feature of field:F2 that semisimple elements form a multiplicative subgroup -- they do not form a multiplicative subgroup for larger field sizes. CAUTION: The subgroup is not the special linear group of degree two! In fact, the special linear group coincides with the whole general linear group. |
As the general affine group:GA(1,q) where | The subgroup comprising translations, i.e., where the scaling factor is . |
Related subgroups
Intermediate subgroups
The subgroup has prime index, hence is maximal, so there are no strictly intermediate subgroups between the subgroup and the whole group.
Smaller subgroups
The subgroup is a group of prime order, so there are no proper nontrivial smaller subgroups contained in it.
Images under quotient maps
Under any quotient map with a nontrivial kernel, the image of the subgroup is trivial. This is because the group is a monolithic group and the subgroup is the unique minimal normal subgroup in it.
Subgroup properties
Invariance under automorphisms and endomorphisms: properties
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | equals all its conjugate subgroups | Yes | Kernel of sign homomorphism; alternating group is always normal in corresponding symmetric group | Also follows from the fact that index two implies normal |
characteristic subgroup | invariant under all automorphisms | Yes | normal Sylow subgroups are characteristic | |
coprime automorphism-invariant subgroup | invariant under all coprime automorphisms, i.e., automorphisms whose order is coprime to that of the group | Yes | ||
cofactorial automorphism-invariant subgroup | invariant under all cofactorial automorphisms, i.e., automorphisms whose order has no prime factors other than those in the group | Yes | ||
subgroup-cofactorial automorphism-invariant subgroup | invariant under all automorphisms whose order has no prime factors other than those in the subgroup | Yes | ||
fully invariant subgroup | contains its image under any endomorphism of whole group | Yes | normal Sylow subgroups are fully invariant |
Resemblance-based properties
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
Sylow subgroup | Yes | |||
normal Sylow subgroup | Yes | |||
Hall subgroup | Yes | |||
normal Hall subgroup | Yes | |||
order-conjugate subgroup | conjugate to all subgroups of the same order | Yes | follows from being a Sylow subgroup, since Sylow implies order-conjugate | |
order-isomorphic subgroup | isomorphic to all subgroups of the same order | Yes | (via order-conjugate, also obvious since has prime order) |
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
verbal subgroup | Yes | |||
isomorph-free subgroup | no other isomorphic subgroup | Yes | ||
order-unique subgroup | no other subgroup of that order | Yes | ||
normal subgroup having no nontrivial homomorphism to its quotient group | Yes | |||
homomorph-containing subgroup | contains any homomorphic image of it in the whole group | Yes | normal Sylow subgroups are homomorph-containing | |
variety-containing subgroup | Yes | |||
quotient-subisomorph-containing subgroup | Yes | |||
image-closed fully invariant subgroup | Yes | (via verbal) | ||
intermediately fully invariant subgroup | Yes | (via variety-containing) | ||
image-closed characteristic subgroup | Yes | |||
intermediately characteristic subgroup | Yes |
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
hereditarily normal subgroup | Yes | |||
transitively normal subgroup | Yes | |||
conjugacy-closed subgroup | No | |||
central factor | No |
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
minimal normal subgroup | nontrivial normal subgroup not containing any other nontrivial normal subgroup | Yes | it is the unique minimal normal subgroup, hence also the socle | |
maximal normal subgroup | proper normal subgroup not contained in any other proper normal subgroup | Yes | it is the unique maximal normal subgroup, hence also the Jacobson radical | |
maximal subgroup | proper subgroup not contained in any other proper subgroup | Yes |
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 three is given, and we need to locate in that the alternating group of degree three. Different ways of constructing/locating this subgroup are given below.
Description | Functions used |
---|---|
SylowSubgroup(G,3) | SylowSubgroup |
DerivedSubgroup(G) | DerivedSubgroup |
FittingSubgroup(G) | FittingSubgroup |
Socle(G) | Socle |
To assign to any of these, do H := followed by that. For instance:
H := SylowSubgroup(G,3);
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(3);; gap> H := AlternatingGroup(3);;
Note that double semicolons have been used to suppress confirmatory output, but you may prefer to use single semicolons.