A good deal of work is being done at present on generalized Frobenius kernels, that is groups which admit a fixed point free automorphism of not necessarily prime order. In this paper we offer a candidate for the title "generalized Frobenius complement." In order to avoid the more difficult problem of studying the groups being acted upon, we consider here only linear groups. We will also assume throughout that the order of the groups are relatively prime to the characteristic of the field so

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... t we have a completely reducible situation. Definition. Let ® be a linear group. Set 0(@) equal to the normal subgroup of ® generated by those elements which have a nonzero fixed point. We call ®/0(®) a generalized Frobenius complement. In §1 we obtain a few simple results about these groups and in §2 we show that these groups come up in a natural way in a problem of D. R. Hughes.