Let G be a group. Define a relation on the set G by bab is a...

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Let G be a group. Define a relation on the set G by bab is a or a ¹. (a) Show that is an equivalence relation on G. (b) Write down the condition for in that case that G is written additively. (c) For the two groups Z6 and Z7, write down the partition given by (remember, by Theorem A2, an equivalence relation on a set partitions the set). (d) Let G be a group of even order (go back to using multiplicative notation here). Show that there is at least one nonidentity element a € G which is its own inverse. (e) What more can you say about the number of such elements in part (d)? Let G be a group. Define a relation on the set G by bab is a or a ¹. (a) Show that is an equivalence relation on G. (b) Write down the condition for in that case that G is written additively. (c) For the two groups Z6 and Z7, write down the partition given by (remember, by Theorem A2, an equivalence relation on a set partitions the set). (d) Let G be a group of even order (go back to using multiplicative notation here). Show that there is at least one nonidentity element a € G which is its own inverse. (e) What more can you say about the number of such elements in part (d)?

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