We prove the rationality of exceptional W-algebras associated with simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case conjecture Kac-Wakimoto. Moreover, we describe $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules compute their characters. also explicit nontrivial action component group on set these modules.

This paper completes the classification of admissible nilpotent orbits of the noncompact simple exceptional real Lie algebras. The author has previoulsly determined such orbits for exceptional real simple Lie algebras of inner type. Here he uses the same techniques, with some modifications, to classify the admissible nilpotent orbits of E6(6) and E6(−26) under their simply connected Lie groups.

If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twisted) rank > 1, other than any F4(q), over a field of known size, a Las Vegas algorithm is given to produce a constructive isomorphism. In view of its timing, this algorithm yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the in...

the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.