aug25 - Math 3124 Thursday, August 25 August 25 Ungraded...

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Math 3124 Thursday, August 25 August 25 Ungraded Homework Problem 4.1 on page 27 With S = { a , b } , the set M ( S ) contains four elements; denote these by π , ρ , σ , τ , defined as follows: π ( a ) = a ρ ( a ) = a σ ( a ) = b τ ( a ) = b π ( b ) = a ρ ( b ) = b σ ( b ) = a τ ( b ) = b (a) Construct the Cayley table for the composition ( ) as an operation on M ( S )= { π , ρ , σ , τ } . (As a start, ρ τ = τ and σ τ = π .) (b) Which is the identity element? (c) Is commutative as an operation on M ( S ) . (d) Which elements of M ( S ) are invertible? (e) Is commutative as an operation on the set of invertible elements in M ( S ) ? The table looks like π ρ σ τ π π π π π ρ π ρ σ τ σ τ σ ρ π τ τ τ τ τ Some sample calculations: to get σ π = τ , we have σ π ( a ) = σ ( a ) = b σ π ( b ) = σ ( a ) = b , consequently σ π ( a ) = τ ( a ) for all a S , which proves that σ π = τ . To get σ σ = ρ , we have σ σ ( a ) = σ ( b ) = a σ σ ( b ) = σ ( a ) = b . From the table, we immediately see that
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This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

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aug25 - Math 3124 Thursday, August 25 August 25 Ungraded...

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