1a) Prove that for any two 3-cycles there exists a such that
1b) Let be the center of . Prove that for all and
1c) Prove that the only nontrivial normal subgroup of is
1d) Give a nontrivial normal subgroup of such that
can anyone help? thanks in advance
Let if is conjugate to then it means .
Therefore, elements in the center are only conjugate to themselves.
We will show that any non-identity element in ( ) are conjugate to some other element.
Let be a cycle, , .
Then .
Therefore if we pick then if .
If then and are the same thing.
But in this special case we can just let where and so .
We have shown that no cycle can be in for .
Let be a non-identity permutation.
Then we can write into a product of disjoint cycles.
Therefore, .
For the moment assume there is one of that is not a transposition, say, by relabing.
Then it means, where .
Let and we see that for .
Therefore, while so since .
We have constructed an element conjugate and not equal to thus .
But we left the case when each is a transposition as an exceptional case.
Try to see what you can come up with to complete the proof for for .
As for the proof should be similar but needs to be done with more care.
If you happen to know that is simple for then that is really helpful here. Accepting this fact we see that is a composition series. By Jordan-Holder theorem this is the unique composition series. Consequently, it must mean that is the only proper non-trivial normal subgroup of .
Choose .1d) Give a nontrivial normal subgroup of such that
can anyone help? thanks in advance
Here is another related problem can you show that and are the only proper non-trivial normal subgroups of ?