What can be said about two groups with isomorphic derived factors?

Question

The third isomorphism theorem states that we can relate an isomorphic relation between two normal subgroups of a group $G$. My question is can we infer anything about the two groups structures itself given that the factor/quotient groups in the derived series of two solvable groups are isomorphic to one another? In other words, if we have two solvable groups $G=\langle g_1,g_2,\ldots, g_n \rangle$ and $H=\langle h_1,h_2,\ldots h_p \rangle$, given that $G^{j}/G^{j+1}$ is isomorphic to $H^{j}/H^{j+1}$, what can be said about $G$ and $H$? (I don't think they are isomorphic but anything less strong be said?)

Answer 1

There is a related concept that is very useful. Given a residually nilpotent group $G$, one can consider the set $$ L(G) = \bigoplus_{i}^{\infty} G_{i}/G_{i+1} $$ where the $G_{i}$ are the terms of the lower central series, defined by $G_{1} = G$, and $G_{i} = [G_{i-1}, G]$ for $i > 1$. Then one can show that $L(G)$ can be given the structure of a Lie ring, where the Lie bracket comes from the group commutator. Several problems on $G$ can be dealt with more efficiently by looking at $L(G)$.

However many non-isomorphic groups $G$ may have isomorphic $L(G)$, so of course not all properties of $G$ can be read off $L(G)$. This is the case for instance for $p$-groups of maximal class $G$ of a fixed order $p^{n}$, with $n > p+1$, that share the same $L(G)$.

Answer 2

If finite groups $G, H$ are metacyclic (i.e. $G'$ and $G/G'$ are cyclic) then your demand implies that their Sylow subroups are isomorphic (since in this case Sylow subroups are cyclic); see M.Hall, The Theory of Groups, Theorem 9.4.3.