Perfect core

In mathematics, in the field of group theory, the '''perfect core''' (or '''perfect radical''') of a group is its largest perfect subgroup.<ref name="Wan1996">{{cite book |last1=Wan |first1=Zhexian |last2=Shi |first2=Sheng-Ming |title=Group Theory in China |date=1996 |publisher=Springer Science & Business Media |isbn=9780792339892 |page=23 |url=https://books.google.com/books?id=VLhj4v7kVxwC&dq=%22perfect+radical%22+group+theory&pg=PA23 |accessdate=1 August 2018 |language=en}}</ref> 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== {{reflist}}

Category:Functional subgroups Category:Group theory Category:Solvable groups