Gruppe

Gruppe is the German noun for “group” and is used in many contexts to denote a collection of elements that are treated as a unit or share a common relation. In German-language usage, the term covers mathematical structures, social aggregates, and organizational units, among others.

In mathematics, a group is an abstract algebraic structure consisting of a set G together with a binary operation • that combines any two elements to form another element of G. The operation is closed, associative, there is an identity element e with e•a = a•e = a for all a in G, and every element a has an inverse b with a•b = b•a = e. If the operation is commutative, the group is called abelian. Common examples include the integers under addition, the nonzero real numbers under multiplication, and the symmetric group on a finite set. Subgroups, homomorphisms, and cosets are central concepts in group theory.

In sociology, a group is a set of individuals who interact and share norms, values, or goals, forming a social unit. Groups vary in size and cohesion; primary groups (family and close friends) emphasize emotional ties, while secondary groups (colleagues, clubs) focus on instrumental purposes. Group dynamics study leadership, conformity, decision making, roles, and groupthink.

In German chemistry, Gruppen is the term for a column of the periodic table; for example, Gruppe 1 contains the alkali metals and Gruppe 17 contains the halogens. In other fields, “Gruppe” may refer to any organizational or categorization unit.

Etymology: the German Gruppe is cognate with the English word group, ultimately from the French groupe.