Jacobson radical
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
Definition
Symbol-free definition
The Jacobson radical (also the Baer radical) of a group is defined in the following equivalent ways:
- As the intersection of all its maximal normal subgroups
- As the subgroup generated by all those elements of the group whose normal closure is a normality-small subgroup
In terms of the intersect-all operator
This property is obtained by applying the intersect-all operator to the property: maximal normal subgroup
View other properties obtained by applying the intersect-all operator
Equivalence of definitions
The equivalence of these definitions follows from Baer's theorem.