111100/Homework Assignment 2
From Drorbn

This assignment is due at class time on Thursday, October 20, 2010.
Solve the following questions
 (Selick)
 What it the least integer for which the symmetric group contains an element of order 18?
 What is the maximal order of an element in ? (That is, of a shuffling of the red cards within a deck of cards?)
 (Selick) Let be a subgroup of index 2 in a group . Show that is normal in .
 Let be a permutation whose cycle decomposition consists of one 5cycle, two 3cycles, and one 2cycle. What is the order of the centralizer of ?
 (Selick) Let be a group of odd order. Show that is not conjugate to unless .
 (Dummit and Foote) Show that if is cyclic then is Abelian.
 (Lang) Prove that if the group of automorphisms of a group is cyclic, then is Abelian.
 (Lang)
 Let be a group and let be a subgroup of finite index. Prove that there is a normal subgroup of , contained in , so that is also finite. (Hint: Let and find a morphism whose kernel is contained in .)
 Let be a group and and be subgroups of . Suppose and . Show that