If two groups are generated by the same number of generators, and the generators have the same orders, are the groups isomorphic? If, say, for two groups G and H, G = < a1, a2, ... , a_n > and H = < b1, b2, ... , b_n >, 
such that |a_i| = |b_i| for all i from 1 to n, 
is G isomorphic to H? 
If so, what is the proof of this?
I know that for n = 1, then H and G are a cyclic group and are isomorphic to Z/|a|Z. Does this extend to larger generating sets?

No; for n=2, we get two distinct groups, where $1$ is the identity
$$G=<a,b | a^2 = 1, aba^{-1}b = 1, b^2=1>$$
$$H=<a,b | a^2 = 1, b^2 = 1>$$
Notice that $G$ is finite - it is $\mathbb{Z_2\times Z_2}$. However, $H$ can be described as the set of words which alternate $a$ and $b$ (with the appropriate multiplication operator) - which is infinite.
