Test elements, generic elements and almost primitivity in free products (Q2809250)

A test element in a group \(G\) is an element \(g\) with the property that if \(f(g)=g\) for an endomorphism \(f\) of \(G\), then \(f\) must be an automorphism.NEWLINENEWLINEA primitive element in a free group is an element which is part of a free basis. In case of an arbitrary group there are three separate different definitions for a primitive element.NEWLINENEWLINELet \(G\) be an arbitrary group.NEWLINENEWLINEAn element \(g\in G\) is a primitive element of type (P1) if a minimal generating system \(X\) of \(G\) exists which contains \(g\).NEWLINENEWLINEAn element \(g\in G\) with \(g\not =\,1\) is a primitive element of type (P2) if \(g\) generates a free factor of \(G\), that is, there is a subgroup \(G_{1}\,\leq \,G\) with \(G\,=\,\langle\,g\, \rangle\ast G_{1}\).NEWLINENEWLINEAn element \(g\in G\) is a primitive element of type (P3) if \(g\) has infinite order and \(g\) generates a free factor of \(G\), that is, there is a subgroup \(G_{1}\,\leq \,G\) with \(G\,=\,\langle\,g\, \rangle\ast G_{1}\).NEWLINENEWLINE(P3) always implies (P2) and (P2) implies (P1). In a free group, the different definitions (P1), (P2), (P3) are equivalent.NEWLINENEWLINELet \(G\) be an arbitrary group. An element \(g\in G\) is called analmost primitive element (according to (P1) or (P2) or (P3) ) if \(g\) is not primitive in \(G\), but it is primitive in every proper subgroup \(K\) of \(G\) which contains \(g\). Therefore, there are three different kinds of almost primitive elements.NEWLINENEWLINELet \(G\) be an arbitrary group and let \(U\) be the variety defined by the set of laws \(V\). An element \(g\in G\) is \(U\)-generic if the following hold:NEWLINENEWLINE(1)\quad \(g\in V(G)\) (the verbal subgroup according the lows in \(V\)) andNEWLINENEWLINE(2)\quad if \(H\) is a group and \(f:H\,\longrightarrow\,G \) with \(g=f(u)\) and \(u\in V(H)\), then \(f\) is surjective.NEWLINENEWLINEIn [\textit{B. Fine} et al., Mat. Contemp. 14, 45--59 (1998; Zbl 0926.20017); \textit{B. Fine} et al., Pac. J. Math. 190, No. 2, 277--297 (1999; Zbl 1009.20031)], the connection between almost primitive and generic elements in free groups was considered. In this paper, the authors study these notions in the case of general free products and in particular to free products of cyclic groups. In distinction to the free groups, in free products it is proved that there are elements which have one of the properties above but not the others. Furthermore, conditions are examined for which the product of primitive elements, almost primitive elements, generic elements and test elements is again such an element.