I think I'm a bit confused about the order of elements in cyclic groups.
If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, or 35. If $x$ is of order 35, then $G$ is cyclic and thus has elements of order 5 and 7.
I don't understand the last part where it says that if $x$ is of order 35, $G$ has elements of order 5 and 7. I'd thought that if $G$ is cyclic then all elements of $G$ would have order 35. Obviously this is wrong, but I don't quite understand why?
Any help would be greatly appreciated!
Best Answer
The order of an elements $g$ in a group $G$ is the smallest number of times that you need to apply the group operation to $g$ to obtain the identity.
Let $G$ be cyclic of order $35$. That means that there is an element $g\in G$ with $g^{35}=e$, and that $g^k\ne e$ for all $1<k<35$. Now, consider $h=g^5$. Then $h^7=(g^5)^7=g^{35}=e$, but $h^k\ne e$ for all $1<k<7$, thus $h$ has order $7$. Similarly, the element $g^7$ has order $5$.
Remark: Cauchy's theorem (which perhaps you did not see yet) states that if $p$ is a prime dividing $|G|$, then $G$ has an element of order $p$. Thus, the only finite groups where all elements except the identity have the same order are $p$-groups, namely groups whose order is a power of a fixed prime $p$. A group of size $35$ is not a $p$-group.