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Unformatted text preview: Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #4 Week #4: October 12th - October 18th. Topics : Composition series, Jordan-Holder’s theorem, solvable groups, higher derived subgroups, nilpotent groups, the Frattini argument, rings, fields. Read : GT [76-84] Problems due Friday, October 16th, at 4:00 pm in Martin Luu’s mailbox: • Problem 1 : As on page 61 in GT, just above Theorem 4.32, make a table of all possible cycle structures for S 5 , and find the cardinalities of all the conjugacy classes in S 5 , together with their parities. • Problem 2 : Let G be a group with | G | = 12 , and assume it contains more than two elements of order three. Show that G is isomorphic to A 4 . • Problem 3 : Let G be a group with | G | = 30 . Show that G is not simple. • Problem 4 : Inside GL n ( C ) , consider the subgroup B consisting of all upper triangular invertible matrices. Compute the derived series of B ....
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This note was uploaded on 05/07/2010 for the course MAT 322 at Princeton.