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hw07 - m i and K = h n i Find(with proof a necessary and...

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Modern Algebra 1 Homework 3.3 Spring 2010 Due February 3 Exercise 10. The set GL 2 ( C ) of invertible 2 × 2 matrices with complex entries can be shown to be a group under matrix multiplication. In fact, the proof given for matrices with real entries can be used, mutatis mutandis , do deal with the case of complex entries. Consider the matrices A = 0 1 - 1 0 and B = 0 i i 0 . Find all the elements in the subgroup Q = h A, B i of GL 2 ( C ). [ Hint: Show that A and B both have finite order and that BA = A 3 B . Use this to prove that every element in Q can be written in the form A i B j , with only a limited number of possibilities for i and j .] Exercise 11. Let m, n Z , H = h m i and
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Unformatted text preview: m i and K = h n i . Find (with proof) a necessary and sufficient condition for us to have H ≤ K . Exercise 12. A subgroup H of a group G is called maximal if H 6 = G and whenever we have H ≤ K ≤ G then K = H or K = G (i.e. there are no subgroups “between” H and G ). Use the result of the previous exercise to determine all the maximal subgroups of Z . Exercise 13. Let H i , i ∈ I , be a collection of subgroups of a group G . Prove that K = \ i ∈ I H i is a subgroup of G . Must the union of subgroups also be a subgroup?...
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