Is a finite group determined by the family of all its 2-generated subgroups?

At the last week I meet my old coauthor, Oleg Verbitsky who proposed me the following question. I think that here should be an easy counterexample, but I am not a pure group theorist and I am usually interested in infinite groups, so I decided to forward the question here. So, here it is:

For a group $G$ and a positive integer $k$ let $Sub_k(G)$ be the family of all subgroups of $G$ generated by its subsets of size at most $k$ and indexed by these subsets, that is $$Sub_k(G)=\{\langle K\rangle_K: K\subset G, |K|\le k\}.$$ Moreover, we shall call two indexed families $\{G_i:i\in I\}$ and $\{G’_{i’}:i’\in I’\}$ of groups isomorphic, if there exists a bijection $\delta:I\to I'$ such that for each $i\in I$ the groups $G_i$ and $G_{\delta(i)}$ are isomorphic.

Now suppose that we have two finite groups $G_1$ and $G_2$ of the equal size $n$. If I remember Oleg’s words right, the groups $G_1$ and $G_2$ are not necessarily isomorphic provided the families $Sub_1(G_1)$ and $Sub_1(G_2)$ are isomorphic, and there is a counterexample of two subgroups of small order. But the isomorphism of the families $Sub_1(G_1)$ and $Sub_1(G_2)$ implies the isomorphism of the groups $G_1$ and $G_2$, provided the groups are abelian. So Oleg asked me: are groups $G_1$ and $G_2$ isomorphic provided the families $Sub_2(G_1)$ and $Sub_2(G_2)$ are isomorphic? He proposed this question to a group theorist, who expects that the answer is negative in general, but positive for the abelian groups.

Thanks.

Solution 1:

No. The groups $$\begin{align} G_{44} &= \langle a,b,c : a^3 = b^9 = c^9 = 1, [b,a]=b^3 c^3, [c,a]=b^3, [c,b]=1 \rangle \\ G_{45} &= \langle a,b,c : a^3 = b^9 = c^9 = 1, [b,a]=c^3 c^3, [c,a]=b^3, [c,b]=1 \rangle \end{align}$$ have isomorphic 2-generated subgroups, but are not themselves isomorphic.

$p$-groups are just a sea of continuous change, so it is not at all surprising to find examples there. Conveniently, it is very easy to find minimal generating sets of $p$-groups, so the condition is more easily checked.

These are the smallest 3-group examples. There are a great many 2-group examples of order 128, including a batch of 6 non-isomorphic groups with isomorphic 2-generated subgroups, but no 2-group examples of smaller order.