## 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, C_{p} has only two subgroups, so there are only two types of orbit, call them X_{1} (with 1 element) and X_{p}, with p elements.

If *p* is odd, then β is given by

- β(X
_{1}) = A_{0}, and - β(X
_{p}) = A_{0}⊕ (sum of all 2-d reps of C_{p}).

If p=2 then

- β(X
_{1}) = A_{0}, and - β(X
_{2}) = A_{0}⊕ A_{1}.

If *n* is not prime, then C_{n} has a subgroup C_{d} for each divisor *d* of *n*. Correspondingly, there is an orbit-type X_{n/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 < ½(n/d).