# Jacobson radical

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

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## 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.