  For a finite p-group G and a bounded below G-spectrum X of finite type mod p,
the G-equivariant Segal conjecture for X asserts that the canonical map X^G -->
X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n.
We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as
well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then
C_{p^n} Segal conjecture holds for X. Similar results hold for weaker forms of
the Segal conjecture, asking only that the canonical map induces an equivalence
in sufficiently high degrees, on homotopy groups with suitable finite
coefficients.
