Transformation near-rings generated by a unit of order three

It is proved that, apart from three exceptions, a finite group G is such that M-0(G) is generated by a unit of order 3 if and only if G is not an elementary 2-group of order congruent to 2 mod 3. The exceptions are C-3, C-2 + C-2, and S-3. The proof divides into four parts. The first (Sec. 3) is where G has at most one element of order 2. The next (Sec. 4) covers nonelementary 2-groups with more such elements. Then, in Sec. 5, we deal with elementary 2-groups. Finally, in Sec. 6, the material of these three parts is brought together to complete the proof.