Conjugate loops: an introduction

In general conjugate of an element does not have any sense in loops. Weintroduce a new class of loops that possess conjugates of their elements. We call such loops conjugate loops. Conjugate loops are loops in which the identity $x(yx^{-1})=(xy)x^{-1}$ holds. We prove that such loops satisfy begin{inparaenum}[(1)] item An IP loop is conjugate $iff$ it is flexible, item Conjugacy is not an equivalence relation in conjugate loops. We also prove several other related results for conjugate loops end{inparaenum}