The diagonal matrices in $GL(n,p)$ can be generated by $n$ elements: simply choose a primitive element $a$ and let $D_i$ be the diagonal matrix with $a$ at the $i$th diagonal entry and 1 everywhere else on the diagonal. Then $D_1,\ldots, D_n$ (semigroup-)generate all diagonal matrices in $GL(n,p)$. It is also know that $GL(n,F)$ is generated by two matrices and the multiplicative group of all $n\times n$ matrices is semigroup-generated by three matrices for every finite field $F$. How about the subgroup of lower triangular matrices?
Question: What is the least number of lower triangular matrices in $GL(n,F)$ that generate the set of all lower triangular matrices in $GL(n,F)$?
Same question about generating as a semigroup can be asked: what is the least number of lower triangular matrices needed to semigroup-generate all of $n\times n$ lower triangular matrices?