order two), and six automorphisms. If Grp were an additive category, then this set E of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0x=x0=0 for all x in the ring, and so z would have to be the zero of E. However, there are no two nonzero elements of E whose product is z, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field by Wedderburn's little theorem, but there is no field with ten elements because every finite field has for its order, the power of a prime. Exact sequences The notion of exact sequence is meaningful in Grp, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in Grp. Grp is a regular category. References Groups Group theory