MAT-3110 Midterm: MATH 3110 App State Spring2015 Test3

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15 Feb 2019
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Name: (15 points) getting things in order. Q = { 1, i, j, k} is the group of quaternions. If not, explain why not. (b) let g be a group with subgroups h and k such that h k g. in addition, suppose that |h| = 2 and |g| = 30. What are the possible orders of k: (10 points) let : g h be a homomorphism between two groups g and h. prove that ker( ) g. [you may not assume that the kernel is a subgroup prove this as well. : (25 points) quotients (a) given: h = {(1), (12)(34), (13)(24), (14)(23)} is a normal subgroup of s4. The size of the set (1234)h is (b) consider where h = h4i = {0, 4, 8, 12, 16}. List all of the cosets (and their contents) of h in z20.

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