How can I show that PGL(2,9) is not isomorphic to S_6?

clovnerie0q

clovnerie0q

Answered question

2022-09-30

How can I show that PGL(2,9) is not isomorphic to S 6 ?
My primary idea is to compare the size of conjugacy classes of two well-chosen elements in these groups. Is there another simpler approach?

Answer & Explanation

aveanoon

aveanoon

Beginner2022-10-01Added 8 answers

Step 1
Let α be a generator of group F 9 . Now if consider the matrix A = diag ( 1 , α ) in PGL(2,9) we have ord ( α ) = 8, since ord ( α ) = 8.
Step 2
But there isn't any element of order 8 in the symmetric group S 6 . So these two groups can not be isomorphic.

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