concrete category of groups

In mathematics, a group is a set of elements combined by an operation that follows specific rules: closure, associativity, an identity element, and inverses for each element. If the operation is also commutative, it's an abelian group.12

The concrete category of groups is a mathematical structure where the objects are groups themselves, and the arrows (or morphisms) are the homomorphisms that preserve the group operations. This concept emerged from the study of abstract groups, which were developed from concrete groups of permutations used to solve polynomial equations. Category theory, introduced by Eilenberg and Mac Lane, provides a framework for studying relationships between different mathematical structures, including groups. Groups are fundamental in abstract algebra and have applications in geometry, physics, chemistry, and computer science.312