# ass9 - ASSIGNMENT 9 MATH235 FALL 2007 Submit by 16:00...

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ASSIGNMENT 9 - MATH235, FALL 2007 Submit by 16:00, Monday, November 19 (1) (a) Let Q 8 be the set of eight elements 1 , ± i, ± j, ± k } in the quaternion ring H , discussed in a previous assignment (so ij = k = - ji etc.). Show that Q 8 is a group. (b) For each of the groups S 3 ,D 4 ,Q 8 do the following: (i) Write their multiplication table; (ii) Find the order of each element of the group; (iii) Find all the subgroups. Which of them are cyclic? (2) (a) Find the order of the permutation σ = ± 1 2 3 4 5 6 7 8 3 1 5 6 2 7 8 4 and write it as a product of cycles. (b) Find a permutation in S 12 of order 60. Is there a permutation of larger order in S 12 ? (3) Let H 1 ,H 2 be subgroups of a group G . Prove that H 1 H 2 is a subgroup of G . (4) Let G be a group and let H 1 ,H 2 be subgroups of G . Prove that if H 1 H 2 is a subgroup then either H 1 H 2 or H 2 H 1 . (5) Let F be a ﬁeld. Prove that SL 2 ( F ) := ‰± a b c d : a,b,c,d F ,ad - bc = 1 ² is a group. If

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ass9 - ASSIGNMENT 9 MATH235 FALL 2007 Submit by 16:00...

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