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Let $G=\{e,a,b,c\}$, $|G|=4$, non-cyclic.

Then what are the orders of the elements?

I'm thinking that because $G$ is a group, then by non-cyclic and Lagrange's theorem its elements (subgroups) can only have order 1 or 2.

But can I decide on anything else than 1 or 2?

mavavilj
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    Since Lagrange theorem gives only the possible orders 1,2,4 and 4 is ruled out by the datum, how could you decide on anything else? Moreover, only the identity, which is unique, ha order 1. Therefore $a,b,c$ have all order 2. – guestDiego Jun 05 '16 at 15:58

1 Answers1

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So every element has order $1$ or $2$, then the element of order $1$ is the identity of course, and all other elements have order $2$, because $1$, $2$ and $4$ are the only divisors of $4$ and the group is not cyclic so there is no element of order $4$.

M. Van
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