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10 Nov 2019
2. Let G be a group and a be a specific element of G. Define (a) If G is any abelian group, what can you say about Ca (for any a E G)? Explain (b) Prove that Ca is a subgroup of G (whether or not G is abelian) 3. Consider the group G = ZeXZ10, which contains 60 elemients all of the form (a,b) where a ⬠26 and b E Z1 Note that this is an additive group (a) What is the order of each of the following elements: (3,5), (2,5), 1), and (5,9)? (b) True or false: if an element g E (a) (i.e., g is in the subgroup generated by a), then (g) is a subgroup of (a) Answer this question at least for this particular group G, but if possible, in general. Justify your answer (c) Is G cyclic? What orders of elements in G are possible? Explain
2. Let G be a group and a be a specific element of G. Define (a) If G is any abelian group, what can you say about Ca (for any a E G)? Explain (b) Prove that Ca is a subgroup of G (whether or not G is abelian) 3. Consider the group G = ZeXZ10, which contains 60 elemients all of the form (a,b) where a ⬠26 and b E Z1 Note that this is an additive group (a) What is the order of each of the following elements: (3,5), (2,5), 1), and (5,9)? (b) True or false: if an element g E (a) (i.e., g is in the subgroup generated by a), then (g) is a subgroup of (a) Answer this question at least for this particular group G, but if possible, in general. Justify your answer (c) Is G cyclic? What orders of elements in G are possible? Explain
Jamar FerryLv2
19 May 2019