Groups

Abstract

Let S be a set. A mapping

$$ S \times S \to S $$

is sometimes called a law of composition (of S into itself). If x, y are elements of S, the image of the pair (x, y) under this mapping is also called their product under the law of composition, and will be denoted by xy. (Sometimes, we also write x · y, and in many cases it is also convenient to use an additive notation, and thus to write x + y. In that case, we call this element the sum of x and y. It is customary to use the notation x + y only when the relation x + y = y + x holds.)