Sylow P-Subgroups, Isomorphisms and Solvable Groups

Related BrainMass Solutions

• 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.

• Simple Groups and Sylow p-Subgroups

  Simple Groups and Sylow p-Subgroups are investigated.

• Fields, Matrix Groups and p-Sylow Subgroups

  150129 Fields, Matrix Groups and p-Sylow Subgroups Please see the attached file for the fully formatted problem.

• Prime simple subgroups

  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 .

• Sylow's Theorem

  b) Assume that a group G has two Sylow p-subgroups K and H. Prove that K and H are isomorphic.

• intersection of normal subgroups is a normal subgroup

  Now recall, that the product of finite number of solvable groups is solvable. Please refer to the attachment for complete solution. Intersections of normal subgroups are exemplified.

• Simple groups

  446717 Simple groups Let G be a group of order n which acts nontrivially on a set of order k. If n>k!, show that G contains a proper normal subgroup. Using the previous result show that if G contains k Sylow p-subgroups, with |G|>k!