Gorenstein, Lyons, and Solomon are currently working on a simplified proof of a classification of a subtype of these groups. For 10 points each:
[10h] Name these groups whose only normal subgroups are the trivial group or the entire group. The monster group is one of the "sporadic" finite groups with this property.
ANSWER: simple groups [or finite simple groups]
[10e] The Feit-Thompson theorem implies that the order of all non-cyclic finite simple groups has this property. Every known perfect number has this property, and two is the only prime number with this property.
ANSWER: even
[10m] The fact that the alternating group on five elements is simple is used to prove this theorem, which states that there is no solution in radicals to a general polynomial of degree five or higher.
ANSWER: Abel-Ruffini theorem [do not accept "Abel's theorem"]
<Andrew Rout , Science - Other - Math Pure>