Groupoids where introduced probably for the first time in Brandt's 1927 paper "Über eine Verallgemeinerung des Gruppenbegriffes", which you can read (if you know German) here. It was conceived from the beginning as a generalization of the concept of a group, namely as a group with partially defined multiplication and inverse maps. There is no real necessary distinction between a group and the corresponding groupoid (see also my comment above). On the second page of his paper, Brandt simply says "Gruppoide vom Rang 1 sind offenbar Gruppen", which translates to "groupoids with one object are evidently groups" (and from the context it is clear that this is meant vice versa). Notice that Brandt defines the rank of a groupoid to be the number of objects (which he calls "Einheiten", i.e. units).

thisseems to be commonly accepted and there is no extra notation. And even more confusingly, $BG$ also denotes (more often, actually) the classifying space of $G$. $\endgroup$ – HeinrichD Nov 8 '16 at 19:37globallythe delooping of a group. $\endgroup$ – Mike Shulman Nov 8 '16 at 22:28naturalcategorical structure formed by groups is a 1-category, whereas the natural categorical structure formed by one-object groupoids is a 2-category, and the two are not the same in any sense. $\endgroup$ – Mike Shulman Nov 9 '16 at 4:10