Group with finite derived subgroup
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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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Contents
Definition
A group is said to have finite derived subgroup if it satisfies the following equivalent conditions:
- Its derived subgroup (i.e., its commutator subgroup with itself) is a finite group.
- The subset of elements of the group that can be written as commutators is a finite subset.
Equivalence of definitions
- The direction (1) implies (2) is obvious.
- The direction (2) implies (1) is proved at finitely many commutators implies finite derived subgroup.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | Yes | follows from the fact that derived subgroup of a subgroup is contained in derived subgroup of the whole group. | If ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | follows from the fact that derived subgroup of a quotient group is the image under the quotient map of the derived subgroup. | Suppose ![]() ![]() ![]() ![]() |
finite direct product-closed group property | Yes | follows from the derived subgroup operation commuting with direct products. | Suppose ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
finite group | the whole group has finitely many elements | (obvious) | any infinite abelian group gives a counterexample. | Finitely generated FZ-group|FULL LIST, MORE INFO |
abelian group | any two elements commute, or equivalently, the derived subgroup is trivial | (obvious) | any finite non-abelian group gives a counterexample. | |FULL LIST, MORE INFO |
FZ-group | the center has finite index, i.e., the inner automorphism group is finite. | FZ implies finite derived subgroup (this result is also called the Schur-Baer theorem) | finite derived subgroup not implies FZ | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
BFC-group | there is a common finite bound on the sizes of all conjugacy classes | this follows from the fact that every conjugacy class is contained in a coset of the derived subgroup | BFC not implies finite derived subgroup | |FULL LIST, MORE INFO |
FC-group | every conjugacy class is finite in size | this follows from the fact that size of conjugacy class is bounded by order of derived subgroup, which in turn is because every conjugacy class is contained in a coset of the derived subgroup. | FC not implies finite derived subgroup | |FULL LIST, MORE INFO |
locally FZ-group | every finitely generated subgroup is a FZ-group | (via FC) | (via FC) | |FULL LIST, MORE INFO |