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 Cp´Cp´Cp, 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(Sn ,n) in terms of the polynomials C(Sk,k), 1 <= k <= n.
E: Check that addition and convolution in fact define a ring structure on the set IF(P) of incidence functions.
E: Let (L,Ù,Ú) denote a lattice and (L, <= ) the corresponding poset. We call fÎIF(L) multiplicative if and only if, for each x,y in L, an order isomorphism[xÙy,xÚy] simeq [xÙy,x]´[xÙy,y]implies thatf(xÙy,xÚy)=f(xÙy,x)f(xÙy,y).
- Prove that the invertible multiplicative fÎIF(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 SX on Xn and on Xni.
harald.fripertinger "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ | UNI-Graz | Institut für Mathematik | UNI-Bayreuth | Lehrstuhl II für Mathematik |
Exercises |