Groups of Prime Power Order as Frobenius-wielandt Complements

It is known that the Sylow subgroups of a Frobenius complement are cyclic or generalized quaternion. In this paper it is shown that there are no restrictions at all on the structure of the Sylow subgroups of the FrobeniusWielandt complements that appear in the well-known Wielandt's generalization of Frobenius' Theorem. Some examples of explicit constructions are also given. 0. Introduction Let G be a finite group acting on a complex vector space M. As in [LP], let N(G, M) be the (normal) subgroup of G generated by those elements of G that fix a nontrivial vector in M. Let N be any normal subgroup of G containing N(G, M). The factor group G/N will be called a FrobeniusWielandt complement (or, shortly, an FW-complement) for G. For an explanation of this name see [E], where these factor groups are shown to be exactly those appearing in Wielandt's generalization of Frobenius' Theorem. In the particular case N = N(G, M) = 1 it is well known that the Sylow p-subgroups of G/N are cyclic or generalized quaternion. On the other hand, one can ask if, given arbitrarily a p-group X, there exists an FW-complement G/N isomorphic to X : here we have the following Theorem. Let X be a finite p-group. Then X is a Frobenius-Wielandt complement: there exists a finite p-group G with a normal subgroup N and a complex G-module M such that N D N(G, M), and G/N is isomorphic to X. §1 is devoted to the proof of the above Theorem, but also contains some results about the p-dimension subgroups of a free group, which turn out to be useful in our setting. Some of these results are known, or at least they belong to the folklore of the theory (e.g., Lemma 1.9, in which the techniques are the same as in [HB, VIII.11.8(c)], or Lemmas 1.11, 1.12 that are essentially contained in [Z, W2]); they are proved here for the convenience of the reader. We note here that the proof of the Theorem rests on the fact that the definition of an FW-complement above is closed under taking factor groups; in other words, homomorphic images of FW-complements are FW-complements. Received by the editors May 17, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D15; Secondary 20F12, 20F14. ©1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page