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.
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:
$$ <1> = \{ 1 \} \\ <a> = \{ 1, a \} \\ <b> = \{ 1,b \} \\ <c> = \{ 1, c \}$$
Obviously, none of these are equal to $\{1,a,b,c \}$, therefore the group is not cyclic.
Is that it?
$\endgroup$ 12 Answers
$\begingroup$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.
$\endgroup$ 2 $\begingroup$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 $<a>$.)
