Group ,  in mathematics, set that has a multiplication that is associative [ a ( bc ) = ( ab ) c for any a, b, c ] and that has an identity element and inverses for all elements of the set. Systems obeying the group laws first appeared in 1770 in Joseph-Louis Lagrange’s studies of permutations of roots of equations; however, the word group was first attached to a system of permutations by Evariste Galois in 1831. It was Heinrich Weber , in 1882, who first gave a purely axiomatic description of a group independently of the nature of its elements. Today, groups are fundamental entities in abstract algebra and are of considerable importance in geometry , physics , and chemistry .