Involutions of finite abelian groups with explicit constructions on finite fields

In this paper, we study properties and constructions of a general family of involutions of finite abelian groups, especially those of finite fields. The involutions we are interested in have the form \(\lambda +g\circ \tau \) $\lambda +g\circ \tau$ , where \(\lambda \) $\lambda$ and \(\tau \) $\tau$ are endomorphisms of a finite abelian group and g is an arbitrary map on this group. We present some involutions explicitly written as polynomials for the special cases of multiplicative and additive groups of finite fields.