Products of subgroups that generate a finite group

Question

Consider the following general problem. There is a finite group $G$ and $H_1,H_2 < G$. Suppose we know that $\langle H_1, H_2 \rangle = G$, i.e. $G$ is generated by $H_1$ and $H_2$. Denote by $n_0$ the minimal $n \in \mathbb{N}$ such that the following holds: $$ G = \underbrace{H_1 H_2 H_1 H_2 \ldots H_1}_{\text{$2n-1$ subgroups}} $$ Problem: How to estimate $n_0$?

This is, definitely, too general.

Question: Do you know about any results of such kind regarding any specific group $G$ and corresponding specific $H_1$ and $H_2$? Maybe matrix groups?

Answer 1

Finite groups for which $n_0=1$ have been studied in various guises by various people:

• Geometric ABA-groups by Higman and McLauglin. This paper connects the study of groups $G$ that factorize as $G=ABA$, with the study of automorphism groups of designs. This is a famous paper that has been cited many times. Note, though, that to make the connection to designs, one needs to place restrictions on the intersection $A\cap B$. The nature of this restriction varies.

• Bruhat decompositions. This is due first to Bruhat, and later to Chevalley, and is a result for reductive linear algebraic groups $G$ (and finite groups of Lie type). The precise statement varies depending on what assumptions you are making about your group but, roughly speaking, it says that one can write $G=BNB$ where $B$ is a Borel subgroup, and $N$ is the normalizer of a maximal split torus.

• Finally, the following paper...

  Alavi, S. Hassan; Praeger, Cheryl E. On triple factorizations of finite groups. J. Group Theory 14 (2011), no. 3, 341–360.

... studies the general question of which groups $G$ can be written as $G=ABA$. The results connect the factorization to the associated permutation representations of $G$ (on the set of cosets of $A$ and $B$).