Order of a cyclic group

Question

I'm struggling to find the order of a cyclic group.

The definition I have is: the order of a group is the number of elements in the group. However, when looking at examples, I get confused.

eg. (134)(25) has order 6. (1254) has order 4. (15)(34) has order 2.

Can anybody explain the definition of order and/or these answers please?

Answer 1

The order of a group $G$ is indeed the number of elements in it. The order of a subgroup $H$ generated by $(12)$ in the symmetric group $G=S_3$, say, is two, because we have $H=\{(1), (12)\}$, with $(12)^2=(1)$. Similarly the subgroup generated by $(15)(34)$ in $S_5$ has only two elements. The cyclic subgroup generated by $(134)(25)$ has order $lcm(2,3)=6$, since the order of $(134)$ is $3$, and the order of $(25)$ is $2$, and $gcd(2,3)=1$, and the two cycles commute. Note that $S_5$ is not cyclic, we need at least two generators, e.g. $(12345)$ and $(12)$.

Answer 2

These are the orders of the elements, i.e. the orders of the cyclic subgroups they generate. Now these permutations all belong to, say, the symmetric group $S_5$, which has order $5!=120$. Note the above-mentioned orders of specific elements are divisors of $120$, in accordance with Lagrange's theorem.