# st1 - ∼ on the set of nonzero real numbers(a a ∼ b if...

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Math 3124 Tuesday, September 13 Sample First Test. Answer All Problems. Please Give Explanations For Your Answers. 1. Let S = {- 1 , 0 , 1 } and deﬁne f : S S by f ( x ) = ( x - 1 ) x ( x + 1 ) for x S , and g : S S by g ( x ) = x 3 . (a) Determine which of f , g is onto. (b) Is f g = g f , where denotes composition? (7 points) 2. Let α = ( 7 3 5 )( 4 7 2 )( 1 6 8 9 )( 6 9 ) . Write α and α - 1 as a product of disjoint cycles. (7 points) 3. Let G = S 5 and T = { 2 , 4 } . Write out all the elements of G T and G ( T ) . Also determine which of these elements are in A 5 . (Recall that G T = { α G | α ( t ) = t for all t T } and G ( T ) = { α G | α ( T ) = T } .) (8 points) 4. Let G be a group, let X G , and let C = { g G | gx = xg } for all x X . Prove that C is a subgroup of G . (8 points) 5. Compute the Cayley table for the symmetry group of the rhombus (parallelogram with all sides equal length, but not square). (7 points) 6. Which of the following deﬁnes an equivalence relation
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Unformatted text preview: ∼ on the set of nonzero real numbers? (a) a ∼ b if and only if a / b ∈ Z . (b) a ∼ b if and only if a / b ∈ Q . (7 points) 7. Let G be the subset of Z 8 consisting of the elements [ 1 ] , [ 3 ] , [ 5 ] , [ 7 ] with the operation ± (modular multiplication). Compute the Cayley table of G . Which elements of G have an inverse? (8 points) 8. Determine the order of the element ± ( 1 2 3 4 5 6 )( 7 8 ) , ² [ 1 ] [ 1 ] [ ] [ 2 ] ³ ´ in S 8 × GL ( 2 , Z 3 ) . (8 points) Test on Tuesday, September 27. Material ﬁrst three chapters, that is up to section 14, approximately . Review session on Sunday September 25 at 4:50 p.m. in McBryde 126....
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