Miscellaneous (Burnside's, simple groups, etc.)

Question 1

What is a simple group?

Answer 1

G is a simple group if its only normal subgroups are the identity and the group itself.

Question 2

Sylow test for nonsimplicity.

Answer 2

Let n be a positive nonprime integer and p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

Question 3

2 Odd test.

Answer 3

An integer of the form 2n where n is odd > 1 is not the order of a simple group.

Question 4

Index Theorem

Answer 4

If G is a finite group and H is a proper subgroup of G such that |G| does not divide |G:H|!, then H contains a nontrivial normal subgroup of G. Thus G is not simple.

Question 5

Embedding theorem.

Answer 5

If a finite non-Abelian simple group G has a subgroup of index n, then G is isomorphic to a subgroup of A_n.

Question 6

Burnside’s Theorem

Answer 6

If G is a finite group of permutations on a set S, then the number of orbits of elements of S under G is
|G|^{-1} sum |fix(phi)| where the sum runs over all permutations phi.