1. References

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 {{harv|Berrick|Robinson|1993}}. The study of imperfect groups apparently began in {{harv|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

• {{Citation | last1=Robinson | first1=Derek John Scott | title=Finiteness conditions and generalized soluble groups. Part 2 | publisher=Springer-Verlag | location=Berlin, New York |mr=0332990 | year=1972}}

• Margarete Poehlmann
• Margarete Rosenberg
• Margaret Erskine (athlete)
• Margaret Erskine of Dun
• Margarete Scheel
• Margarete Schick
• Margarete Schnate
• Margarete Schramböck
• Margaret E. Slade
• Margarete Slezak
• Margaret E. Sloan
• Margarete Steinborn
• Margarete Susman
• Margaret E. Thompson
• Margarete Thüring
• Margaret Etim
• Margaret Evans (journalist)
• Margaret Evelyn de Arias
• Margaret Evelyn Simpson
• Margaret Eve Oppenheimer
• Margarete von Bayern
• Margaret E. Winslow
• Margarete Wittkowski
• Margarete Zuelzer
• Margaret Fairweather