abstract. let g be a connected algebraic group. an unrefinable chain of g is a chain
of subgroups g = g0 > g1 > · · · > gt = 1, where each gi is a maximal connected
subgroup of gi−1 . we introduce the notion of the length (respectively, depth) of g,
defined as the maximal (respectively, minimal) length of such a chain. working over an
algebraically closed field, we calculate the length of a connected group g in terms of the
dimension of its unipotent radical ru (g) and the dimension of a borel subgroup b of
the reductive quotient g/ru (g). in particular, a simple algebraic group of rank r has
length dim b + r, which gives a natural extension of a theorem of solomon and turull on
finite quasisimple groups of lie type. we then deduce that the length of any connected
algebraic group g exceeds 12 dim g.
we also study the depth of simple algebraic groups. in characteristic zero, we show
that the depth of such a group is at most 6 (this bound is sharp). in the positive
characteristic setting, we calculate the exact depth of each exceptional algebraic group
and we prove that the depth of a classical group (over a fixed algebraically closed field
of positive characteristic) tends to infinity with the rank of the group.
finally we study the chain difference of an algebraic group, which is the difference
between its length and its depth. in particular we prove that, for any connected algebraic
group g, the dimension of g/r(g) is bounded above in terms of the chain difference of
g.