GROUP OPERATIONS ON A GROUP
Z Chen · Journal of Southwest China Normal University · 1985
Letbe a word of a group G. Tehn xoy=f(x, y) is an algebraic operation of G.Theorem 1 Suppose that the group G has a finite exponent n, then G becomes a group G° with respect to the operation o :where rs≡1 (mod n), and k is a fixed element of G. Moreover G is isomorphicto G°.Theorem 2 Suppose that G is a nilpotent group of class z, then G becomes a group G° with respect to xoy=f(x, y) if and only ifLet G° is a group}.Suppose that O1, O2∈Ω, xO1y=f1(x, y) . The word f(x, y)o2 is derived by replacing the original operations in f1(x, y) with o2. Define the product o = o1o2 as xoy=f1(x, y)o2.Theorem 3 Ω is a monoid.Ω is called the induced semigroup of G.Theorem 4 G is abelian if and only if the induced semigroup of G is the identity group.Theorem 5 The induced semigroup of a (free) nilpotent group of class 2 is isomorphic to a sub-semigroup of multiplicative semigroup of the number