The Sylow Theorems

Let S be a non empty 1-sorted structure, let E be a set, let A be an action of the carrier of S on E, and let s be an element of S. We introduce A a s as a synonym of A(s). Let S be a non empty 1-sorted structure, let E be a set, let A be an action of the carrier of S on E, and let s be an element of S. Then A a s is a function from E into E. Let S be a unital non empty groupoid, let E be a set, and let A be an action of the carrier of S on E. We say that A is left-operation if and only if: (Def. 1) A a (1S) = idE and for all elements s1, s2 of S holds A a (s1 · s2) = (A a s1) · (A a s2). Let S be a unital non empty groupoid and let E be a set. Note that there exists an action of the carrier of S on E which is left-operation. Let S be a unital non empty groupoid and let E be a set. A left operation of S on E is a left-operation action of the carrier of S on E.