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 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 that
f(xÙy,xÚy)=f(xÙy,x)f(xÙy,y).
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
last changed: January 19, 2005

 Exercises