For a finite group that needs n generator to construct, is the number of elements of the group the same as the product of the numbers of elements each of these generators alone can generate?
If so,is it also true that for two finite groups, if they have the same minimum number of generators needed to construct and that each generator alone can generate the same number of elements corresponding to another group, then the two groups are isomorphic?
For example, if I two groups that need 2 generators,and one of the generator generates 3 elements(including e) and another one generates 6 elements, then (3-1)(6-1)=10 is the number of elements of the group and the two groups are isomorphic to each other?
I just started learning group theory a few days ago, so hope you don't mind if I actually ask something stupid.
Briefly , My attempt is that the set generated by a generator is a cyclic group, and that the direct product of those cyclic groups should be isomorphic to the original group itself.
Best Answer
$S_n$ can be generated by $\sigma = (12)$ and $\tau = (12\dots n)$. The order of $\sigma$ is $2$ and the order of $\tau$ is $n$. However, the number of elements in $S_n$ is $n!$.