Imperfect group

Nowadays, Imperfect group is a topic that is becoming more and more relevant in our society. Over the years, we have seen how Imperfect group has impacted our lives in various ways, from the way we communicate, to the way we carry out our daily activities. There is no doubt that Imperfect group has generated a great impact both individually and collectively, and that is why more and more people are seeking information and knowledge on this topic. In this article, we will thoroughly explore Imperfect group and analyze its influence on different aspects of our life.

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).[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


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