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.
References
References
- Wan, Zhexian; Shi, Sheng-Ming (1996). Group Theory in China. Springer Science & Business Media. p. 23. ISBN 9780792339892. Retrieved 1 August 2018.