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As Nick points out, you don't have to impose any extra conditions to get a positive answer. This suggests a formulation based primarioly on the Tits axioms for a $BN$-pair, refined somewhat for your …

As Mark Wildon points out, there is a closely related question already on MO here. Both questions are answered in a relatively elementary style by the old method of Frobenius, which is presented in …

Geoff is on the right track about the way Suzuki groups occur naturally as subgroups of others. But in view of the incomplete formulation of the original question, and the string of comments followi …

All I can suggest is a partial answer. In the narrower case when you look at a Borel subgroup of the finite special linear group, there should be no outer automorphisms arising from conjugation withi …

Two sources come to mind, though there may be more recent ones. These are both available online now. 1) A concise exposition of central extensions is given by Steinberg in Section 7 of his 1967-68 …

Some useful references are given below, readily available online (with access to JSTOR). Weisfeiler's short paper was published just before his disappearance while hiking in Chile in January 1985, b …

The question is somewhat loosely stated, leading to various answers and comments which are at cross-purposes. Some specific families of finite groups are mentioned, but the list seems to be left o …

To focus just on the question of giving a presentation of a finite general linear group over the prime field (or other finite field), leaving aside the fuzzy connection with representations of finite …

As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der …

To clarify a bit what Igor refers to, the answer to the question asked is that "primitive degrees" for a finite Coxeter group such as a Weyl group are usually abbreviated to "degrees", meaning the uni …

Taken in isolation the question is reasonable, but it's important to add the explicit journal reference: J. Algebra 105 (1987), 542-573. This paper is rather technical and necessarily heavy on notat …

What is meant by "finite group of Lie type" needs to be made precise. But at least the simple groups of Lie type in characteristic $p$ with a cyclic Sylow $p$-subgroup are easy to specify: these are …

The Annals paper by Roggenkamp and Scott was certainly a landmark in the ongoing study of the isomorphism problem for integral group rings of finite groups, which apparently goes back to the thesis wo …

It may help to have some explicit bibliographic references, though some items are by now out of print and may be difficult to locate even through libraries. First, the 1966 paper by Gorenstein is lo …

This kind of question has a long history (pre-computer), but it's always been difficult to get much intuition about solvable groups just from looking at a big list. In the case of simple groups the o …