USNA Pure Mathematics Seminar

The talks for the academic year 2001-2002 are held Wednesday in Chauvenet 201 at 3:45 pm unless otherwise stated.



Antonia W. Bluher




A natural isomorphism between A6 and PSL2(9)


Abstract: Let F = F3m, where m is odd, and let a be a nonzero element of F. Arasu and Player proved through an indirect argument that the polynomial

(x6 + x)2 - a2 = (x6 + x + a)(x6 + x - a)
always has exactly one root or exactly four roots in F. We decided to study the polynomial x6 + x + a in order to obtain a more direct proof. It turns out that the polynomial is closely related to x10 + a x + 1. Specifically, there are 10 ways to partition the 6 roots of x6 + x + a into two sets of size 3, i.e. 10 ways to factor x6 + x + a into two cubics, and these 10 factorizations correspond naturally to the 10 roots of x10 + a x + 1. Using this, we can give a direct proof of the result of Arasu and Player. Further, by studying the splitting fields and Galois groups of these two polynomials (but now taking a to be transcendental), we obtain a natural isomorphism between the alternating group A6 (which arises as the Galois group of x6 + x + a) and the projective linear group PSL2(9) (which arises as the Galois group of x10 + a x + 1). While these two groups were known to be isomorphic, a natural realization of the isomorphism was not previously known.


Time: noon, Monday February 25, 2002


Reception at 11:50am in the common room on the 3rd floor of Chauvenet Hall.