Problems with normal subgroups

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• Normal Subgroups of a Group

  Subgroups of a Group Normal Subgroups of a Group Suppose that N and M are two normal subgroups of G and that N intersection M = (e).

• Intersection of two normal subgroups is a normal subgroup.

  Normal Subgroups of a Group Show that the intersection of two normal subgroups of G is a normal subgroup of G.

• Sylow theorem

  361467 Normal Sylow Theorem Let G be a group of order 48. By the 1st Sylow theorem G has a Sylow 2-subgroup and a Sylow 3-subgroup. Suppose none of these are normal. Determine the number of Sylow 2-subgroups and Sylow 3-subgroups that G can have.

• Normal Subgroups of a Group

  Subgroups of a Group Normal Subgroups of a Group Subgroups of a Group Normal Subgroups of a Group Suppose H is the only subgroup of order O(H) in the finite group

• a full description of alternating group A(4), discussion of its subgroups (normal, sylow, cyclic), and their interrelationships.

  34831 discussion of its subgroups (normal, sylow, cyclic) Need a full description of alternating group A(4), discussion of its subgroups (normal, sylow, cyclic), and their interrelationships.

• Simple groups

  , show that G contains a proper normal subgroup. Using the previous result show that if G contains k Sylow p-subgroups, with |G|>k!, then G is not simple. See the attachment. This solution helps with problems involving simple groups.

• Sylow 5-subgroup

  Therefore, G has a unique Sylow 3-subgroup with 2 nonidentity elements of order 3. Now the number of Sylow 2-subgroups must satisfy and It follows that and G has elements of order 2.

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

  423560 Proof that There Are No Simple Groups of Order 56 or 148 A simple group is a group with nontrivial normal subgroups. Show that there are no simple groups of order 148 or of order 56. First suppose |G| = 56 = 2^3 * 7.

• 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 .