Group with finitely many homomorphisms to any finite group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Contents
Definition
Equivalent definitions in tabular format
No. | Shorthand | A group is termed a group with finitely many homomorphisms to any finite group if ... |
---|---|---|
1 | finitely many homomorphisms to any finite group | for any finite group , there are only finitely many homomorphisms from to . |
2 | finitely many surjective homomorphisms to any finite group | for any finite group , there are only finitely many surjective homomorphisms from to . |
3 | finitely many normal subgroups of fixed finite index | for any natural number , has only finitely many normal subgroups of finite index with the index equal to . |
4 | finitely many normal subgroups of bounded finite index | for any natural number , has only finitely many normal subgroups of finite index with the index at most . |
5 | finitely many subgroups of fixed finite index | for any natural number , has only finitely many subgroups of finite index with the index equal to . |
6 | finitely many subgroups of bounded finite index | for any natural number , has only finitely many subgroups of finite index with the index at most . |
7 | intersection of normal subgroups of fixed finite index has finite index | for any natural number , the intersection of all the normal subgroups of of index equal to is also a normal subgroup of finite index in . |
8 | intersection of subgroups of fixed finite index has finite index | for any natural number , the intersection of all the subgroups of of index equal to is also a subgroup of finite index in . |
9 | intersection of normal subgroups of bounded finite index has finite index | for any natural number , the intersection of all the normal subgroups of of index at most is also a normal subgroup of finite index in . |
10 | intersection of subgroups of bounded finite index has finite index | for any natural number , the intersection of all the subgroups of of index at most is also a subgroup of finite index in . |
Equivalence of definitions
Further information: equivalence of definitions of group with finitely many homomorphisms to any finite group
The nontrivial ingredients in the equivalences are Poincare's theorem (which asserts that a subgroup of finite index contains a normal subgroup of finite index at most ) and that index satisfies intersection inequality, which yields that the intersection of finitely many subgroups of finite index again has finite index. We also need to use the fact that a subgroup of finite index can be contained in only finitely many intermediate subgroups.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
finite group | ||||
finitely generated group | finitely generated implies finitely many homomorphisms to any finite group | finitely many homomorphisms to any finite group not implies finitely generated | |FULL LIST, MORE INFO | |
simple group | simple implies finitely many homomorphisms to any finite group | (obvious) | |FULL LIST, MORE INFO | |
group of finite composition length | finite composition length implies finitely many homomorphisms to any finite group |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
group in which every subgroup of finite index has finitely many automorphic subgroups | finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups |