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In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point (the origin), where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.