Given a group $G$, we denote by $T(G)$ the subgroup generated by all (maximal) normal abelian subgroups of $G$.

Let define the series $(T_i(G))$ by $T_0(G)=1$ and $T_{i+1}(G)/T_i(G)=T(G/T_i(G)$, and $n(G)$ to be the smallest integer such that $T_n(G)=G$.

1. Can one produce finite p-groups $G$ with arbitrary large n(G)?

2. If $G$ is a finite p-group, what can one say in general about the index of $T(G)$ in $G$.

3. I found that the nilpotency class of $T(G)$ (assuming just that $G$ is finite), can not exceed $s$, where $s$ is the cardinal of the set of conjugacy class sizes in $G$. In fact if $1< n_1 <...<n_s$ are the conjugacy class sizes of $G$, we define $H_i$ to be the subgroup generated by the elements with conj class sizes $\leq n_i$, we use a result of Mann and Isaacs to show that $T(G)$ is a stability group of the series $H_i$ and the claim follows from a well known result of Kaloujnin.

Is this result trivial ?