Perfect group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition
Symbol-free definition
A group is said to be perfect if it equals its own commutator subgroup.
Definition with symbols
A group is said to be perfect if .
In terms of the fixed-point operator
The property of being perfect is obtained by applying the fixed-point operator to a subgroup-defining function, namely the commutator subgroup.
Property theory
Direct products
This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
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A direct product of perfect groups is a perfect group. This follows from the fact that the commutator subgroup of a direct product is the direct product of the commutator subgroups.
Quotients
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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Any quotient of a perfect group is perfect. Thus, the property of being perfect is a quotient-hereditary subgroup property.
Subgroup-generation
The subgroup generated by any two perfect subgroups of a group is also a perfect subgroup.