Large characteristic subgroups with modular subgroup lattice

Invariant systems of representatives, or The cost of symmetry

“…set of orders of all elements) of the quotient group G/H is contained in the spectrum of G/N [KlMi15]; -if an algebra G (associative or Lie) over a field contains a solvable ideal of finite codimension, then G contains an invariant with respect to all automorphisms solvable (of the same derived length) ideal of finite codimension [KhM08]. The list can be extended, see, e.g., [Vd00], [KhM07b], [KhM08], [KlMe09], [KhKMM09], [MSh12], [KlMi15], [Fr18], [dGT18a], [dGT18b], [dGT19a], [dGT19b], and references therein. The assertions of this kind are called sometimes ([KlMi15], [Fr18]) Khukhro-Makarenko type theorems after one of such results [KhM07a] (see also [KlMe09]) including as special cases the first four of the listed facts.…”