FINITE GROUPS OF RANK TWO WHICH DO NOT INVOLVE Qd(p)

Let p > 3 be a prime. We show that if G is a finite group with p-rank equal to two, then G involves Qd(p) if and only if G p'-involves Qd(p). This allows us to use a version of Glauberman's ZJ-theorem to give a more direct construction of finite group actions on mod-p homotopy spheres. We give an example to illustrate that the above conclusion does not hold for p <= 3.