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 Évariste Galois (Galois, Évariste) 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.

in chemistry, a set of chemical elements in the same vertical column of the periodic table. The elements in a group have similarities in the electronic configuration of their atoms, and thus they exhibit somewhat related physical and chemical properties.

The periodic table has eight main groups: 1, 2, 13, 14, 15, 16, 17, and 18 (previously numbered Ia, IIa, IIIa, IVa, Va, VIa, VIIa, and 0, respectively). Each group consists of elements that have similar electronic structures characterized by completely filled inner electron shells and by a number of electrons in their outermost shells equal to the group number. Ten other groups—3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 (previously numbered IIIb, IVb, Vb, VIb, VIIb, VIII, Ib, and IIb, with group VIII comprising groups 8, 9, and 10)—found only in Periods 4 to 7 of the table, are composed of elements of the transition series. With these elements the number of outermost electrons does not necessarily correspond to the group number.