Finitely generated implies finitely many homomorphisms to any finite group
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated group) must also satisfy the second group property (i.e., group with finitely many homomorphisms to any finite group)
View all group property implications | View all group property non-implications
Get more facts about finitely generated group|Get more facts about group with finitely many homomorphisms to any finite group
Suppose is a finitely generated group. Then, is a group with finitely many homomorphisms to any finite group. More explicitly, if is a finite group, then the set of homomorphisms from to is finite.
In fact, we have the following explicit upper bound on the number of homomorphisms: if the minimum size of generating set for is and the order of is , then the number of homomorphisms from to is .
Equality holds in the case that is a finitely generated free group of rank , though it may also hold in other cases. More generally, equality holds if admits a quotient that is free of rank in the subvariety of the variety of groups generated by . For instance, if is abelian, it suffics for have a surjective homomorphism to a free abelian group of rank .
|Statement for group with finitely many homomorphisms to any finite group||Corollary for finitely generated group|
|finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups||finitely generated implies every subgroup of finite index has finitely many automorphic subgroups|
|residually finite and finitely many homomorphisms to any finite group implies Hopfian||finitely generated and residually finite implies Hopfian|
To prove: There are at most homomorphisms from to .
Proof: By Fact (1), to specify a homomorphism from to , it suffices to specify the restriction to the generating set . Thus, the total number of homomorphisms from to is bounded by the number of set maps from to . By set theory, this number is .
Comment on equality
Note that the bound is achieved if and only if every set map from to extends to a homomorphism from to .