Exercise 6, § 5

Let n be an integer > 0 and let Γ(Z/nZ) be the gamma algebra of the Z-module Z/nZ (Exercise 2). Show that for every p > 0, Γ_p(Z/nZ) is isomorphic to the quotient of Z by the subgroup generated by the integers n^k((k, p - k)), where 1 ≤ k ≤ p. Show that if p and n are distinct primes, then Γ_p(Z/nZ) is cyclic of order n. Show that if n is even, then Γ_2(Z/nZ) is cyclic of order 2n.