It has often been noted that modular arithmetic provides a rich source of supply of groups. Indeed, a remarkable theorem asserts that any finite commutative group can be found by a sufficiently diligent search through the multiplicative groups and subgroups of modular arithmetic! During the last few years two articles in this area have appeared in Mathematics Teaching. In the first [1], Tim Brand drew attention to the fact that such a multiplicative group can have an identity element other than 1. (For example, 8 is the identity of the group {2,4,8} under multiplication mod 14.) More recently [2] Geoff Saltmarsh described an ingenious way of finding the identity and put forward an interesting conjecture about these groups.