## 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.

**Speaker**:
# Antonia W. Bluher

##
NSA

** Title:**

#
A natural isomorphism between A_{6} and PSL_{2}(9)

**Abstract**:
Let F = F_{3m},
where m is odd, and let a be a nonzero element of F.
Arasu and
Player proved through an indirect argument that the polynomial

(x^{6} + x)^{2} - a^{2}
= (x^{6} + x + a)(x^{6} + x - a)
always has exactly one root or exactly four roots in F.
We decided to study the
polynomial x^{6} + x + a in order to obtain a more direct proof.
It turns out
that the polynomial is closely related to x^{10} + a x + 1.
Specifically, there
are 10 ways to partition the 6 roots of x^{6} + x + a into
two sets of size 3,
i.e. 10 ways to factor x^{6} + x + a into two cubics, and
these 10 factorizations
correspond naturally to the 10 roots of x^{10} + 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 A_{6} (which arises as the Galois group of x^{6} + x + a)
and the projective
linear group PSL_{2}(9) (which arises as the Galois group of
x^{10} + 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.