Combinations and Permutations Episode 31: S/T

Showing a group of 42 elements is not simple follows directly from Sylow Theorems:
a. There must be a subgroup of order 7.
b. The number of subgroups of order 7 must be congruent to 1 (mod 7), and must divide 42. Thus, there must be exactly one subgroup of order 7.
c. Because there is exactly one such subgroup, it must be normal. Therefore, the group is not simple.

Showing a group of order 72 is not simple, on the other hand, is quite a bit harder. Showing a group of order 90 is not simple is a bear.

I forget who said it. It was a tangent.
The job search has entered Year Two – Year One ended with an adjunct offer, which was a nice change from “sorry, it’s just not the right fit.” I’m a little hampered by geographical concerns, but I’m hopeful that the improved economy leads to more hiring – the campus down the street might have several openings.