Imperfect group

{{about|the mathematical topic|groups named Imperfect|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 {{harv|Berrick|Robinson|1993}}. The study of imperfect groups apparently began in {{harv|Robinson|1972}}.<ref>That this is the first such investigation is indicated in {{harv|Berrick|Robinson|1993}}</ref>

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 ''Sym''<sub>2</sub> 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''|<sup>2</sup>).

==References== {{refimprove|date=February 2008}} <references/> * {{Citation | last1=Berrick | first1=A. J. | last2=Robinson | first2=Derek John Scott | title=Imperfect groups | doi=10.1016/0022-4049(93)90008-H |mr=1233309 | year=1993 | journal=Journal of Pure and Applied Algebra | issn=0022-4049 | volume=88 | issue=1 | pages=3–22| doi-access= }} *{{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}}

{{DEFAULTSORT:Imperfect Group}} Category:Properties of groups