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

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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 = { a n : n Z } (take a 0 = 1), and define a m · a n = a m + n and φ ( a ) = a = a 1 . 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 ϕ : S G is any set map, then there is a unique group homomorphism f : A ( S ) G such that f | S = ϕ . (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
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