normal subgroups of the symmetric groups
For , is the only proper nontrivial normal subgroup of .
This is essentially a corollary of the simplicity of the alternating groups for . Let be normal. Clearly . But is simple, so or . In the first case, either , or else also contains an odd (http://planetmath.org/SignatureOfAPermutation) permutation, in which case . In the second case, either or else consists solely of one or more odd permutations in addition to . But if contains two distinct odd permutations, and , then either or , and both and are even (http://planetmath.org/SignatureOfAPermutation), contradicting the assumption that contains only odd nontrivial permutations. Thus must be of order , consisting of a single odd permutation of order 2 together with the identity.
It is easy to see, however, that such a subgroup cannot be normal. An odd permutation of order , , has as its cycle decomposition one or more (an odd number, in fact, though this does not matter here) of disjoint transpositions. Suppose wlog that is one of these transpositions. Then takes to and thus is neither nor . So this group is not normal. ∎
If , is the trivial group, so it has no nontrivial [normal] subgroups.
If , , the unique group on elements, so it has no nontrivial [normal] subgroups.
If , has one nontrivial proper normal subgroup, namely the group generated by .
is the most interesting case for . The arguments in the theorem above do not apply since is not simple. Recall that a normal subgroup must be a union of conjugacy classes of elements, and that conjugate elements in have the same cycle type. If we examine the sizes of the various conjugacy classes of , we get
A subgroup of must be of order , or (the factors of ). Since each subgroup must contain , it is easy to see that the only possible nontrivial normal subgroups have orders and . The order subgroup is , while the order subgroup is . is obviously normal, being of index , and one can easily check that is also normal in . So these are the only two nontrivial proper normal subgroups of .
|Title||normal subgroups of the symmetric groups|
|Date of creation||2013-03-22 17:31:38|
|Last modified on||2013-03-22 17:31:38|
|Last modified by||rm50 (10146)|