Definability of group theoretic notions

We consider logical definability of the group-theoretic notions of simplicity, nilpotency and solvability on the class of finite groups. On one hand, we show that these notions are definable by sentences of deterministic transitive closure logic DTC. These results are based on known group-theoretic results. On the other hand, we prove that simplicity, nilpotency and the normal closure of a subset of a group are not definable by single sentences of first order logic FO. In addition, we show that ...