MAT_305-t1

MAT_305-t1 - ρ ρ-1(c Is ρ in A 8(d Calculate ρ 13 4...

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MAT 305 Fall’07 Test 1 SHOW ALL YOUR WORK TO GET PARTIAL CREDIT. 1. Find the order of each element in the indicated group: (a) 6 in Z 15 (b) ( 2 7 3 )( 1 2 4 ) in S 8 , (c) 11 in U(14) and (d) 286 in Z 462 2. Let G = <a> be a cyclic group with order |G| = 15 (a) List all the generators of G (b) List all the elements of G of order 5 (c) Explain why G has no element of order 6 (d) Explain why if ϕ : G Z 4 is a homomorphism then ϕ (x) = 0, 2200 x in G. 3. Let ρ = ( 1 5 6 2 )( 1 2 )( 3 4 7 8 )( 3 8 ) in S 10 , (a) Find the order | ρ | (b) Find the inverse of
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Unformatted text preview: ρ , ρ-1 (c) Is ρ in A 8 ? (d) Calculate ρ 13 4. Determine if any of the following pairs are isomorphic groups.( JUSTIFY ) (a) A 3 and Z 3 (b) U(12) and Z 4 (c) U(14) and Z 6 (d) U(18) and Z 18 5. In S 4 consider the subgroup H = A 4 ∩ D 4 . (a) List all the elements of H (b) Find all the distinct left cosets of H in A 4 (c) Find all the distinct left cosets of H in D 4 (d) H is isomorphic to which group we are familiar with? Explain. Each problem is worth 20points....
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