 The abstract concept of group in full generality is defined not in terms of a set of numbers but rather as an arbitrary set equipped with a binary operation \(xy\), a unary inverse \(x^{-1}\) of that operation, and a unit \(e\) satisfying certain equations characteristic of groups. One striking novelty with groups not encountered in everyday elementary algebra is that their multiplication need not be abelian: \(xy\) and \(yx\) can be different! For example the group \(S_3\) of the six possible permutations of three things is not abelian, as can be seen by exchanging adjacent pairs of letters in the word dan. If you exchange the two letters on the left before the two on the right you get adn and then and, but if you perform these exchanges in the other order you get dna and then nda instead of and. Likewise the group of 43,252,003,274,489,856,000 operations on Rubik’s cube and the infinite group \(SO(3)\) of rotations of the sphere are not abelian, though the infinite group \(SO(2)\) of rotations of the circle is abelian. Quaternion multiplication and matrix multiplication is also noncommutative. Abelian groups are often called additive groups and their group operation is referred to as addition \(x+y\) rather than multiplication \(xy\).