If $H$ is a normal subgroup of $S_n$ containing a transposition, show that $H=S_n$.

Question

Where are you seeing this? Most group theory textbook should covers this. — Lynnx, Oct 17 '20 at 05:21

score 1 · Accepted Answer · answered Oct 17 '20 at 05:31

1

Every two transpositions are conjugate. So if a normal subgroup contains one of them it contains all of them and so the whole symmetric group is equal to the normal subgroup.

answered Oct 17 '20 at 05:31

markvs

19,653

What do you mean by "every two transpositions are conjugate"? – chickenwing72 Oct 17 '20 at 05:45
@chickenwing72 More generally, any elements of $S_n$ with the same cycle structure are conjugate. – Oct 17 '20 at 05:48

If $H$ is a normal subgroup of $S_n$ containing a transposition, show that $H=S_n$.

1 Answers1