Permutation representations of the cyclic group

For cyclic groups, the morphism β : Ω(G) → R(G) (taking Burnside ring to representation ring) is injective (is this the only type of group where it is?).

If p is prime, Cp has only two subgroups, so there are only two types of orbit, call them X1 (with 1 element) and Xp, with p elements.

If p is odd, then β is given by

  • β(X1) = A0, and
  • β(Xp) = A0 ⊕ (sum of all 2-d reps of Cp).

If p=2 then

  • β(X1) = A0, and
  • β(X2) = A0 ⊕ A1.

If n is not prime, then Cn has a subgroup Cd for each divisor d of n. Correspondingly, there is an orbit-type Xn/d with n/d elements. And

  • \(\beta(X_{n/d}) = A_0 \oplus \sum E_{rd}\) if n/d is odd
  • \(\beta(X_{n/d}) = A_0 \oplus A_1 \oplus \sum E_{rd}\) if n/d is even,

where the sum is over r with 1 ≤ r < (n/d).