11-1100/Homework Assignment 2
This assignment is due at class time on Thursday, October 20, 2010.
Solve the following questions
- What it the least integer n for which the symmetric group Sn contains an element of order 18?
- What is the maximal order of an element in S26? (That is, of a shuffling of the red cards within a deck of cards?)
- (Selick) Let H be a subgroup of index 2 in a group G. Show that H is normal in G.
- 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 σ?
- (Selick) Let G be a group of odd order. Show that x is not conjugate to x − 1 unless x = e.
- (Dummit and Foote) Show that if G / Z(G) is cyclic then G is Abelian.
- (Lang) Prove that if the group of automorphisms of a group G is cyclic, then G is Abelian.
- Let G be a group and let H be a subgroup of finite index. Prove that there is a normal subgroup N of G, contained in H, so that (G:N) is also finite. (Hint: Let (G:H) = n and find a morphism whose kernel is contained in H.)
- Let G be a group and H1 and H2 be subgroups of G. Suppose and . Show that