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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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

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.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online