10-1100/Homework Assignment 2
This assignment is due at class time on Thursday, October 21, 2010.
Solve the following questions
- 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 5-cycle, two 3-cycles, and one 2-cycle. 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.
- 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