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I just want to make sure I have the right idea here.

The Statement of the Problem:

Prove that the four-group $\{1,a,b,c \}$ is not cyclic.

My Answer:

As far as I can tell, this is the Klein four-group and I just need to check the subgroups generated by each element. If any of them is the entire group, then it is cyclic; otherwise, it is not cyclic. Well:

\begin{align}\langle1\rangle &= \{ 1 \} \\ \langle a\rangle &= \{ 1, a \} \\ \langle b\rangle &= \{ 1,b \} \\ \langle c\rangle &= \{ 1, c \}\end{align}

Obviously, none of these are equal to $\{1,a,b,c \}$, therefore the group is not cyclic.

Is that it?

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    $\begingroup$ Yes, your proof is perfectly good. $\endgroup$
    – pjs36
    Oct 14, 2015 at 2:18

2 Answers 2

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A cyclic group of order $4$ has an element of order four, and the Klein four group doesn't: every element is of order $2$. Alternatively, a cyclic group of order four has a unique subgroup of order $2$, and the Klein four group has three (distinct) subgroups of order two.

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    $\begingroup$ The question was not "is the Klein four-group cyclic" or "how can I prove that the Klein four-group is not cyclic",. The question was about the correctness of the OP's proof. $\endgroup$
    – bof
    Oct 14, 2015 at 2:33
  • $\begingroup$ @bof Thanks, I didn't notice that. $\endgroup$
    – Pedro
    Oct 14, 2015 at 2:54
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Yes, your proof is correct.

(LaTeX note: angle brackets are coded as \langle a\rangle instead of <a>. The former becomes $\langle a\rangle$ and the latter becomes $<a>$.)

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    $\begingroup$ (I would've made that a comment rather than an answer, but it answers the question, so…) $\endgroup$ Oct 30, 2015 at 15:57

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