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).
^That this is the first such investigation is indicated in (Berrick & Robinson 1993)
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Perfect consonances: unisons and octaves perfect fourths and perfect fifths Imperfect consonances: major thirds and minor sixths minor thirds and major sixths...
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reduced for every field extension F/k. (see below) Otherwise, k is called imperfect. In particular, all fields of characteristic zero and all finite fields...
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the present Imperfect (imparfait) Pluperfect (plus-que-parfait): literally "more than perfect", formed with an auxiliary verb in the imperfect Simple past...
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