Sylow p-Subgroups, Conjugacy and Abelian Groups

Related BrainMass Solutions

• Prime simple subgroups

  (a) If p = q, then G is abelian, and contains a subgroup of order p (by Cauchy), which is necessarily normal. If p =/= q, say p < q, then there is a unique Sylow q-subgroup Q, which must be normal.

• Abstract Algebra : Abelian Groups, Cauchy's Theorem, LaGrange's Theorem and Sylow Subgroups

  Abelian Groups, Cauchy's Theorem, LaGrange's Theorem and Sylow Subgroups are investigated.

• 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, Isomorphisms and Solvable Groups

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• 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.

• 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.

• Group Theory : Conjugacy, Cayley Table, Subgroups and Quotient Groups

  Identify the elements of the quotient group G/H , and determine its isomorphism type. Please see the attached file for the complete solution. Conjugacy, cayley table, subgroups and quotient groups are investigated and discussed.