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

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.

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?

• Yes, your proof is perfectly good. Oct 14, 2015 at 2:18

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.

• 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.
– bof
Oct 14, 2015 at 2:33
• @bof Thanks, I didn't notice that.
– Pedro
Oct 14, 2015 at 2:54

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