Imperfect group

This article is about the mathematical topic. For groups named Imperfect, see Imperfect (disambiguation).

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).

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

• 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