These transformations are examples of affine involutions. Extraspecial groups often occur in centralizers of involutions. These are not the only pre-calculus involutions. The other groups arise as fixed points of involutions. Co has 4 conjugacy classes of involutions. Don Zagier's short proof based on involutions have appeared. Groups of symplectic type appear in centralizers of involutions of groups of GF(2)-type. Other elementary involutions are useful in solving functional equations. Note that all of these operations are involutions. Scramble and descramble is done by repeating four basic functions (involutions). The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups. For example, formula 87 can be generated by two elements or by involutions. Koecher exhibits explicit involutions of simple Euclidean Jordan algebras directly (see below). The remaining cases are constructed uniformly by Koecher using involutions of simple Euclidean Jordan algebras.