Sublattices generated by root differences (Q403611)

Let \(\Phi\) be an irreducible root system, let \(\mathbb{Z}\Phi\) be the corresponding root lattice. For a sublattice \(L\subset \mathbb{Z}\Phi\) denote as \(t(L)\) the exponents of the torsion subgroup of \(\mathbb{Z}\Phi/L\). Let NEWLINE\[NEWLINET(\Phi)=\{t(L):\mathrm{ \(L\) a sublattice of \(\mathbb{Z}\Phi\) generated by root differences}\}.NEWLINE\]NEWLINE Put \(t(\Phi)=\max T(\Phi)\).NEWLINENEWLINEThe main result of the paper are the following equalities NEWLINE\[NEWLINEt(G_2)=12,\,\, t(F_4)=68,\,\, t(E_6)=124,\,\, t(E_7)=388,\,\, t(E_8)=1312.NEWLINE\]NEWLINENEWLINENEWLINEThis result is of interest since \(t(\Phi)\) appears in a theorem of \textit{M. W. Liebeck} and \textit{G. M. Seitz} [``On the subgroup structure of exceptional groups of Lie type'', Trans. Am. Math. Soc. 350, No. 9, 3409--3482 (1998; Zbl 0905.20031)] concerning liftings of embeddings from finite groups of Lie type to algebraic groups.