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Hi I need an example of a cyclic group with exactly 6 generators, thank you.

user278815
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If it has $6$ generators, and since $e$ is never a generator (unless it's the trivial group, which it's not), then the group you are looking for must have at least $7$ elements. How many cyclic groups with $7$ elements do you know? (hint: essentially just one.) For that single most immediate candidate as an answer, is it true that each of its elements other than $e$ is a generator? (Hint: the order of an element in a cyclic group is easy to compute. What do you know about it?)

Ittay Weiss
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  • Must it have precisely 7 elements though? That seems to be false. Certainly the group with 7 elements works, but what of a group with 14 elements? What of others? – JMoravitz Oct 10 '15 at 06:20
  • A4 (alternating subgroup of S4) is what Ittay is talking about. – Yunus Syed Oct 10 '15 at 06:28
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    thanks @JMoravitz I meant to write "at least 7 elements" (corrected now). My point was that a very minimalistic approach leads to a candidate which actually solves the question. – Ittay Weiss Oct 10 '15 at 06:43
  • @JMoravitz 7, 9, 14, 18. – bof Oct 10 '15 at 06:56
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    @YunusSyed I don't see anything about A4 in what Ittay wrote. His answer was entirely about cyclic groups, and A4 isn't cyclic. – Andreas Blass Oct 10 '15 at 07:38