It is well-known that if in a group the usual multiplication \(pq\) is replaced by \(pa^{-1}q\) for some fixed element \(a\) of the group, the resulting structure is again a group, isomorphic to the original one, which is this group with \(a=e\), the unit element. The authors study in some detail these “groups based at \(a\)”. They compare especially the groups based at two different elements of the original group. A typical theorem is [Theorem 2.2] that an \(a\)-based subgroup of the group based at \(a\) is simply a coset \(Ua\) or \(aU\) of an ordinary subgroup \(U\) of the original group. Again an ordinary subgroup of the original group is \(a\)-based Abelian [according to the obvious definition] if it is ordinarily Abelian, and it then is also \(b\)-based Abelian for every other element \(b\) [Proposition 3.2, which is based, like several other results, on the second author’s paper in Arch. Math. 47, 522-528 (1986; Zbl 0594.20073)]. There are more such definitions and results, too many for reproduction here.