In this work the author lets \(\Phi\) be an irreducible root system, with Coxeter group \(W\). He considers subsets of \(\Phi\) which are abelian, meaning that no two roots in the set have sum in \(\Phi \cup \{ 0 \}\). He classifies all maximal abelian sets (i.e., abelian sets properly contained in no other) up to the action of \(W\): for each \(W\)-orbit of maximal abelian sets we provide an explicit representative \(X\), identify the (setwise) stabilizer \(W_X\) of \(X\) in \(W\), and decompose \(X\) into \(W_X\)-orbits.

Abelian sets of roots are closely related to abelian unipotent subgroups of simple algebraic groups, and thus to abelian \(p\)-subgroups of finite groups of Lie type over fields of characteristic \(p\). Parts of the work presented here have been used to confirm the \(p\)-rank of \(E_8(p^n)\), and (somewhat unexpectedly) to obtain for the first time the \(2\)-ranks of the Monster and Baby Monster sporadic groups, together with the double cover of the latter.

Root systems of classical type are dealt with quickly here; the vast majority of the present work concerns those of exceptional type. In these root systems the author introduces the notion of a radical set; such a set corresponds to a subgroup of a simple algebraic group lying in the unipotent radical of a certain maximal parabolic subgroup. The classification of radical maximal abelian sets for the larger root systems of exceptional type presents an interesting challenge; it is accomplished by converting the problem to that of classifying certain graphs modulo a particular equivalence relation.