MAT-3110 Midterm: MATH 3110 App State Spring2009 Test2 answer key

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15 Feb 2019
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March 25th, 2009: (16 points) either prove g is a group or explain why it is not a group. (a) g = 5z = {m z | m is divisible by 5} (under addition). G is a group since it is a subgroup of z (under addition). To show it is a subgroup notice that: 0 = 5 0 g so g 6= o(cid:14) . 5(m + n ) g. thus g is closed under addition: m, n g implies that m = 5m and n = 5n for some m , n z. So m + n = 5m + 5n : m g implies that m = 5m for some m z. So m = 5m = 5( m ) g. thus g contains inverses for each of its elements. Therefore, by the subgroup test, g is a subgroup of z.

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