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This assignment is due at class time on Thursday, October 20, 2010.

===Solve the following questions===

# (Selick)
## What it the least integer <math>n</math> for which the symmetric group <math>S_n</math> contains an element of order 18?
## What is the maximal order of an element in <math>S_{26}</math>? (That is, of a shuffling of the red cards within a deck of cards?)
# (Selick) Let <math>H</math> be a subgroup of index 2 in a group <math>G</math>. Show that <math>H</math> is normal in <math>G</math>.
# Let <math>\sigma\in S_{20}</math> 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 <math>C_{S_{20}}(\sigma)</math> of <math>\sigma</math>?
# (Selick) Let <math>G</math> be a group of odd order. Show that <math>x</math> is not conjugate to <math>x^{-1}</math> unless <math>x=e</math>.
# (Dummit and Foote) Show that if <math>G/Z(G)</math> is cyclic then <math>G</math> is Abelian.
# (Lang) Prove that if the group of automorphisms of a group <math>G</math> is cyclic, then <math>G</math> is Abelian.
# (Lang)
## Let <math>G</math> be a group and let <math>H</math> be a subgroup of finite index. Prove that there is a normal subgroup <math>N</math> of <math>G</math>, contained in <math>H</math>, so that <math>(G:N)</math> is also finite. (Hint: Let <math>(G:H)=n</math> and find a morphism <math>G\to S_n</math> whose kernel is contained in <math>H</math>.)
## Let <math>G</math> be a group and <math>H_1</math> and <math>H_2</math> be subgroups of <math>G</math>. Suppose <math>(G:H_1)<\infty</math> and <math>(G:H_2)<\infty</math>. Show that <math>(G:H_1\cap H_2)<\infty</math>

11-1100/Homework Assignment 2 - Revision history

Drorbn at 12:52, 13 October 2011

Drorbn at 22:10, 10 October 2011