| Last Math Table: | 2/28/06: | |
The outer automorphism of S_6 by Noam Elkies, Harvard University |
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Abstract: The symmetric group S6 of
permutations of 6 objects
is the only symmetric group with an outer
automorphism.
We outline two approaches to the existence and construction
of such an outer automorphism of S6:
one via the 15 simple and 15 triple transpositions,
the other via the transitive index-6 subgroup PSL2(Z/5Z) of S6.
This outer automorphism can be regarded as the seed from which grow about half of the sporadic simple groups, starting with the Mathieu groups M12 and M24. It also enters unexpectedly into a classical description of the space of configurations of six distinct 1-dimensional subspaces of a 2-dimensional vector space, and motivates an open problem of the late Walter Feit on conjugate sextics Time permitting, we'll at least hint at some of these manifestations of the outer automorphism of S6. |