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

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

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