Imperfect group

In mathematics, in the area of algebra known as group theory, an imperfect group is a group with no nontrivial perfect quotients. Some of their basic properties were established in (Berrick & Robinson 1993). The study of imperfect groups apparently began in (Robinson 1972).^[1]

The class of imperfect groups is closed under extension and quotient groups, but not under subgroups. If G is a group, N, M are normal subgroups with G/N and G/M imperfect, then G/(N∩M) is imperfect, showing that the class of imperfect groups is a formation. The (restricted or unrestricted) direct product of imperfect groups is imperfect.

Every solvable group is imperfect. Finite symmetric groups are also imperfect. The general linear groups PGL(2,q) are imperfect for q an odd prime power. For any group H, the wreath product H wr Sym₂ of H with the symmetric group on two points is imperfect. In particular, every group can be embedded as a two-step subnormal subgroup of an imperfect group of roughly the same cardinality (2|H|²).

References

1. That this is the first such investigation is indicated in (Berrick & Robinson 1993)

• Berrick, A. J.; Robinson, Derek John Scott (1993), "Imperfect groups", Journal of Pure and Applied Algebra 88 (1): 3–22, doi:10.1016/0022-4049(93)90008-H, ISSN 0022-4049, MR 1233309
• Robinson, Derek John Scott (1972), Finiteness conditions and generalized soluble groups. Part 2, Berlin, New York: Springer-Verlag, MR 0332990