Friday, 7 December 2018

abstract algebra - Prove that, operatornameAut(S3)simeqS3simeqoperatornameInn(S3)

Actually I have solved the part "Inn(S3)S3" by using the result-
"Let G be a group. Then GZ(G)Inn(G), where Z(G)={gGgx=xgxG}"



So I put G=S3 and hence Z(S3)={e}, where e is identity permutation.
And thus S3Z(S3)S3
Finally, using the above result, we get, Inn(S3)S3.
But how to prove Aut(S3)S3. Although, One thing I have notice that, to prove Aut(S3)S3, it is enough to show that Inn(S3)Aut(S3). But I am stuck with it also.
Can anyone suggest me a clear and rigorous wayout?
Thanks for your help in advance.

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