Fuchs' problem for p-groups

Abstract

Which groups can be the group of units in a ring? This open question, posed by László Fuchs in 1960, has been studied by the authors and others with a variety of restrictions on either the class of groups or the class of rings under consideration. In the present work, we investigate Fuchs' problem for the class of p-groups. Ditor provided a solution in the finite, odd-primary case in 1970. Our first main result is that a finite 2-group G is the group of units of a ring of odd characteristic if and only if G is of the form C8t×∏i=1kC2nisi, where t and si are non-negative integers and 2ni+1 is a Fermat prime for all i. We also determine the finite abelian 2-groups of rank at most 2 that are realizable over the class of rings of characteristic 2, and we give some results concerning the realizability of 2-groups in characteristic 0 and 2n. Finally, we show that the only almost cyclic 2-groups which appear as the group of units in a ring are C2,C4,C8,Cq−1 (q a Fermat prime), C2×C2n(n≥1), D8, and Q8. From this list we obtain the p-groups with periodic cohomology which arise as the group of units in a ring.