PGL(2,9) is not isomorphic to S6

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Statement

The groups projective general linear group:PGL(2,9) (defined as the projective general linear group of degree two over field:F9) is not isomorphic to symmetric group:S6.

This non-isomorphism is important to keep in mind because PSL(2,9) is isomorphic to A6, and thus both these groups have a subgroup of index two that is isomorphic (namely, A6 in S6 and PSL(2,9) in PGL(2,9)). There is also a third group, Mathieu group:M10, that has a subgroup of index two isomorphic to alternating group:A6 that is not a direct factor. For more on this perspective, see supergroups of alternating group:A6.

Related facts

Opposite facts

Proof

Proof using conjugacy class sizes

Using the element structure of projective general linear group:PGL(2,9), we see that the conjugacy class size statistics are:

1, 36, 45, 72, 72, 72, 72, 80, 90, 90, 90

On the other hand, using element structure of symmetric group:S6, we see that the conjugacy class size statistics are:

1, 15, 15, 40, 40, 45, 90, 90, 120, 120, 144

The conjugacy class sizes do not match up, so the groups are not isomorphic.

To show non-isomorphism, it is not necessary to compute the full conjugacy class size statistics of both groups, but we need to compute enough to show that the conjugacy class size statistics cannot match up.