Hopfian group
Encyclopedia
In mathematics
, a Hopfian group is a group
G for which every epimorphism
is an isomorphism
. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotient
s. A group G is co-Hopfian if every
monomorphism
is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Hopfian group is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G for which every epimorphism
- G → G
is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
s. A group G is co-Hopfian if every
monomorphism
- G → G
is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
.
Examples of Hopfian groups
- Every finite groupFinite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
, by an elementary counting argument. - More generally, every polycyclic-by-finite group.
- Any finitely-generated free groupFree groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
. - The group Q of rationalsRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
. - Any finitely generated residually finite groupResidually finite groupIn the mathematical field of group theory, a group G is residually finite or finitely approximable if for every nontrivial element g in G there is a homomorphism h from G to a finite group, such thath \neq 1.\,...
. - Any torsion-free word-hyperbolic group.
Examples of non-Hopfian groups
- Quasicyclic groups.
- The group R of real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. - The Baumslag-Solitar group B(2,3).