HW9_Solutions

HW9_Solutions - Math 541 Solution to HW#9 Assignment Prove...

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Math 541 Solution to HW #9 Assignment: Prove that if G is a simple group of size less than 60, then G is cyclic of prime size. Recall from class the following lemmas: Lemma P: If | G | = p with p a prime, then G is a simple group. Lemma Z: If | G | = p a with p a prime and a an integer greater than 1, then G is not a simple group. Lemma S: If | G | is not a prime, and G has a singleton conjugacy class not equal to { e } , then G is not simple. Lemma r!: If G has a conjugacy class of size r > 1 and | G | > r !, then G is not simple. Super Lemma r!: If G has a conjugacy class of size r > 1 with | G | < r !, and | G | does not divide r !, then G is not simple. As a consequence of the above lemmas, we can add the following: Lemma PQ: If | G | = pq with p and q both primes, then G is not a simple group. Proof : Seeking a contradiction, let G be a group of order pq that is simple. By Lemma S, G has no singleton conjugacy class besides the identity. Since the size of any conjugacy class of

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HW9_Solutions - Math 541 Solution to HW#9 Assignment Prove...

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