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In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup.[1] Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup. The perfect core is also the point where the transfinite derived series stabilizes for any group.
A group whose perfect core is trivial is termed a hypoabelian group. Every solvable group is hypoabelian, and so is every free group. More generally, every residually solvable group is hypoabelian.
The quotient of a group G by its perfect core is hypoabelian, and is called the hypoabelianization of G.
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