Variety with no nontrivial perfect member is solvable

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Statement

Consider a subvariety of the variety of groups (i.e., a collection of groups closed under taking subgroups, quotients, and arbitrary direct products). If no nontrivial group in this variety is perfect, i.e., if every nontrivial member of the variety is distinct from its commutator subgroup, then the variety is a solvable variety of groups.

References

Textbook references

  • Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, More info, Page 23, Theorem 1.8.1 (Section 1.8) (formal statement with proof)