Fields, Matrix Groups and p-Sylow Subgroups

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• Proof that There Are No Simple Groups of Order 56 or 148

  Now G can only be simple if it has 7 Sylow-2 subgroups and 8 Sylow-7 subgroups. We know from Sylow's second theorem that all Sylow-p subgroups are conjugate. In particular, all eight Sylow-7 subgroups must be conjugate.

• Sylow p-Subgroups, Conjugacy and Abelian Groups

  We also note ( ) because has factor and has factor . Thus is a sylow-p subgroup of . Sylow p-Subgroups, Conjugacy and Abelian Groups are investigated. The solution is detailed and well presented.

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  226532 Prime simple subgroups a) Show that if |G|= pq, where p and q are prime, then G is not simple. b) Show that the only simple groups of order less than 36 are of prime order.

• Prove that a group of order 108 must have a normal subgroup of order 9 or 27.

  Hence the possibility for 4(four) 3-Sylow subgroups is is normal in and , where and are any two 3-Sylow subgroups of .

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