Unformatted text preview: G = R { } , a * b =  ab  [10 pts] (b) G = Z , a * b = a + b + ab [10 pts] 3. Find the right cosets of the subgroup H = h (1 , 1) i in Z 2 × Z 4 . [10 pts] 4. Let G be an abelian group (possibly inﬁnite) and let the set T = { a ∈ G  a m = e for some m > 1 } . (a) Prove T is a subgroup of G . [10 pts] (b) Prove that G/T has no element  other than its identity element  of ﬁnite order. [15 pts] 5. Suppose  G  = 10 and G has a subgroup of order 2 which is not normal. Prove that G is isomorphic to a subgroup of S 5 . [15 pts] 6. Determine all groups (up to isomorphism) of order 4225 = 5 2 · 13 2 . Justify your answer completely. [20 pts]...
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 Spring '11
 Johnson
 Math, Algebra, pts, Subgroup, point values, answer booklet

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