Generate All Permutations of Sn from An and 1 Odd Permutation

[gravenewworld]

say you have the alternating group An for some permutation group Sn. If you are given An and then 1 odd permutation, must you be able to generate all of Sn? I tried it for S3 and I multiplied all the even perms in S3 by only 1 element that wasn't in A3 and was able to generate all of S3. Does this hold for any n?

 

[Galileo]

Yes, that is generally true.
Note that in any group multiplication on the left by an element in the group is a bijection.
ax=b \iff x=a^{-1}b
Use this to prove it for the general case S_n ,n\geq 2
The case n=1 is special, since the A1=S1.

 

[gravenewworld]

Alright thanks a lot galileo. I just wanted to be sure of that fact before I brought it up in my presentation that I have to give.