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).
Famous quotes containing the words imperfect and/or group:
“The imperfect is our paradise.
Note that, in this bitterness, delight,
Since the imperfect is so hot in us,
Lies in flawed words and stubborn sounds.”
—Wallace Stevens (1879–1955)
“A little group of willful men, representing no opinion but their own, have rendered the great government of the United States helpless and contemptible.”
—Woodrow Wilson (1856–1924)