Abstract Which groups can be the group of units in a ring? This open question, posed by Laszlo Fuchs in 1960, has been studied by the authors and others with… Click to show full abstract
Abstract Which groups can be the group of units in a ring? This open question, posed by Laszlo 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 C 8 t × ∏ i = 1 k C 2 n i s i , where t and s i are non-negative integers and 2 n i + 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 2 n . Finally, we show that the only almost cyclic 2-groups which appear as the group of units in a ring are C 2 , C 4 , C 8 , C q − 1 (q a Fermat prime), C 2 × C 2 n ( n ≥ 1 ) , D 8 , and Q 8 . From this list we obtain the p-groups with periodic cohomology which arise as the group of units in a ring.
               
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