s09_mthsc851_hw07

s09_mthsc851_hw07 - MTHSC 851 (Abstract Algebra) Dr....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 7 Due Thursday March 26th, 2009 (1) Verify that if S = {1} and N = {1, 2, 3, . . . }, then (N, +) is a free semigroup on S, with j(1) = 1. (2) Prove that if a free semigroup X exists on S, it is unique up to isomorphism. (3) If S = {a} let F = {an : n Z} (take a0 = 1), and define am an = am+n and (a) = a = a1 . Show that F is a free group on S. Thus a free group on a single "generator" a can be taken to be the infinite cyclic group a . (4) (a) Let S be a set. The group with presentation (S, R) where R = {[s, t] | s, t S} is called the free abelian group on S denote it by A(S). Prove that A(S) has the following universal property: if G is any abelian group and : S G is any set map, then there is a unique group homomorphism f : A(S) G such that f |S = . (b) Deduce that if A is a free abelian group on a set of cardinality n, then A Z Z Z (n factors). = (5) Show that every nonidentity element in a free group F has infinite order. (6) Let F be a free group and let N be the subgroup generated by the set {xn | x F, n a fixed integer}. Show that N F . (7) Let (F, ) be a free group on a set S. (a) If |S| 2, show that F is not abelian. (b) For any T S, show that there exists a normal subgroup N F such that (F/N, |T ) is a free group on T . (c) Show that ( (T ) , |T ) is a free group on T . (8) For a positive integer n, let Cn be the category of nilpotent groups of class at most n. Prove or disprove that free objects always exist in Cn . (9) Let F be a free object on S, and F a free object on S in a concrete category C. Prove that F and F are equivalent. ...
View Full Document

This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.

Ask a homework question - tutors are online