Relation between Minimum number of generator and number of elements?

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.

• The situation cannot be as simple as your last paragraph suggests, for that would mean all finite groups are abelian (evidently false). Sep 13 '18 at 0:31
• @hardmath, you are right, I thought something wrong about the direct product. The operation is definitely not as simple. Sep 13 '18 at 1:00

$S_n$ 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 $n!$.
• No problem! This is a good question to ask. The order of the group being an upper bound for the product of generators of course depends on the generating set. For example, if $G$ is a group then the set $G$ generates itself and so the product of the orders of all elements of $G$ is probably much larger than the order of $G$. Given a finite group, can you ever pick a minimal generating set such that the product of the orders of the generators (there has to be an easier way to say that) is larger than the order of the group? I need to think about this some more. Sep 13 '18 at 1:56