Periodic $\lambda$-rings and exponents of finite groups

A $\lambda$-ring is said to be $n$-periodic if its Adams operators satisfy the relation $\psi^{i+n}=\psi^i$ for each $i$. The quotient by the radical of the free periodic $\lambda$-ring generated by one element is described. Using this description, the order of a finite group is shown to divide the group's exponent to the power equal to the dimension of an arbitrary faithful complex representation.