321final - c(10 points(7 Let G be a group H a commutative...

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MATH 321 Final Examination Your name and surname: Jan 16, 2001 Your nickname (if applicable): 9:00–12:00 Your best signature: PARK 1 Question Score 5 /10 points 10 /5 points 1 /5 points 6 /10 points 11 /5 points 2 /5 points 7 /10 points 12 /10 points 3 /5 points 8 /7 points 13 /15 points 4 /5 points 9 /8 points Total /100 points (1) Give an example of a fact that you have known long before taking MATH 321 but which you recognized in MATH 321 to be a piece of information about groups. (5 points) (2) Denote the residue class of a Z modulo 18 by ¯ a , modulo 24 by ˜ a . Is the “mapping” ϕ : Z 18 -→ Z 24 , ¯ a 7→ ˜ a a group homomorphism? (5 points) 1
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(3) Are Z × 9 and Z × 18 isomorphic groups? (5 points) (4) Can there exist a group G and an element a of G such that o ( a 3 ) = 20 and o ( a 5 ) = 9? Either give an example of such a pair G , a , or prove that this is impossible. (5 points) 2
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(5) Let F = h f i be a cyclic group of order 4 and T = h t i a cyclic group of order 10. Find all group homomorphisms from F into T . (10 points) 3
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(6) If a , b , c Z 11 and G := { g GL(2 , Z 11 ) : det g ∈ { 4 , 5 , a, b, c }} is a group, what are a , b and
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Unformatted text preview: c ? (10 points) (7) Let G be a group, H a commutative group, ϕ : G → H a homomorphism onto H . Let N be a subgroup of G that contains Ker ϕ . Show that N is a normal subgroup of G . (10 points) 4 (8) Let H be a subgroup of a group G . Suppose g ∈ G and o ( g ) = n ∈ N . If g m ∈ H and if m and n are relatively prime, prove that g ∈ H . (7 points) (9) Let G be a group, N ≤ G , A ⊆ G . If N £ G , does AN = NA have to hold true? (8 points) 5 (10) Give an example of a Feld di±erent from Q , R , C , Z p ( p prime). (5 points) (11) Characterize all rings in which the identity x 2-y 2 = ( x + y )( x-y ) is valid. (5 points) 6 (12) Let A be an ideal of R . Is it true that Mat 2 ( R ) / Mat 2 ( A ) ∼ = Mat 2 ( R/A )? (10 points) 7 (13) Let D be a principal ideal domain, a, b relatively prime elements of D . Is D/abD isomorphic to D/aD ⊕ D/bD ? (15 points) 8...
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This note was uploaded on 11/08/2011 for the course MATH 321 taught by Professor Ergenc during the Spring '11 term at Boğaziçi University.

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321final - c(10 points(7 Let G be a group H a commutative...

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