Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms stabilizing a flag $V=V_1\supset V_2\supset\cdots\supset V_1$ is called a parabolic subgroup of $GL(V)$. I am interested in the subgroup $Q$ of $P$ consisting of all automorphisms $h$ such that the induced automorphism $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the identity for every $i$. My question is:
Is this subgroup $Q$ named somewhere yet? If not, can you recommend a name?
Similarly, I am also interested in the subgroup $T$ of $P$ consisting of automorphisms $h$ such that the group of induced automorphisms $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is the symmetric group (on a basis of $V_i/V_{i+1}$). Is this subgroup $T$ named somewhere yet? If not, can you recommend a name? Is there anyway to realize that $\overline{h}: V_i/V_{i+1}\rightarrow V_i/V_{i+1}$ is a symmetric group without looking a specific basis of $V_i/V_{i+1}$)?