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# Publication
- Dr Tim Burness

Burness, TC, Liebeck, MW & Shalev, A, 2019, ‘The length and depth of algebraic groups’. *Mathematische Zeitschrift*, vol 291., pp. 741-760

finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds 1/2 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 with soluble radical R(G), the dimension of G/R(G) is bounded above in terms

of the chain difference of G.