Exercises

Exercises

E: Check the equation for the cycle index of S₄.

E: Verify the details of locally finite partial orders.

E: Evaluate the cycle indicator polynomial of the action of the group C_p´C_p´C_p, p being an odd prime, on itself by left multiplication. Evaluate also the cycle indicator polynomial of the action of the nonabelian group of order p³ on itself by left multiplication.

E: Express the cycle indicator C(S_n ,n) in terms of the polynomials C(S_k,k), 1 <= k <= n.

E: Check that addition and convolution in fact define a ring structure on the set I_F(P) of incidence functions.

E: Let (L,Ù,Ú) denote a lattice and (L, <= ) the corresponding poset. We call fÎI_F(L) multiplicative if and only if, for each x,y in L, an order isomorphism

[xÙy,xÚy] simeq [xÙy,x]´[xÙy,y]

implies that

f(xÙy,xÚy)=f(xÙy,x)f(xÙy,y).

• Prove that the invertible multiplicative fÎI_F(L) form a group.
• Show that the zeta function (and hence also the Moebius function) is multiplicative.
• Verify the description of the interval [d,n] and the Moebius function on the lattice of divisors.

E: Prove the details in the examples.

E: Evaluate the characters of the natural actions of S_X on Xⁿ and on Xⁿ_i.

Exercises