I am currently working on number theory and I have come across a question asking for non-trivial subgroups of a given group. the question states that G = Z*73, I know this is equal to all values of 73 which are inversable so that is 1..72.
I understand that a subgroup must satisfy closure, identity, inverses and be associative.
i know that trivial subgroups are the ones that any group can have, when the subset is equal to the identity element or G itself.
but when trying to work out non-trivial ones, as it is a large number 73, I don't know how I would go about working out subgroups of it?
any infomration would be appreciated!
Best Answer
By Lagrange, every subgroup of $U(73)$ has order $d\mid \phi(73)=72$ Consider the subgroup $U=\langle 2\rangle $, generated by $2$. Since $2^9\equiv 1\bmod 73$, $U$ is a non-trivial subgroup of $U(73)$, having order $d=9$.