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

Question

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?

Answer 1

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.

Answer 2

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 $<a>$.)