The socle of the center of a group algebra

Let A be a finite-dimensional algebra over an algebraically closed field F. We consider the socle soc(Z(A)) of its center Z(A), which is known to be an ideal of Z(A). The principal question treated in this thesis is under which conditions soc(Z(A)) is even an ideal in the entire algebra A. Moreover, we study the analogous problem for the Jacobson radical J(Z(A)) of the center as well as the Reynolds ideal of A. Our main focus lies on the case that A is the group algebra FG of a finite group G over F. We first classify the finite p-groups G for which soc(ZFG) is an ideal in FG. The main focus of this thesis, however, lies on the analysis of the structure of arbitrary finite groups G for which soc(ZFG) is an ideal in FG. We give a complete characterization of these groups a particular case. In the general setting, we state a conjecture on the structure of G, which we prove in a special case. We also investigate the main problem for symmetric algebras, with a particular focus on symmetric local algebras. We find examples of symmetric local algebras A of minimal dimension in which the Jacobson radical J(Z(A)) or the socle soc(Z(A)) are not ideals, respectively. Moreover, these properties are investigated for quantum complete intersection algebras as well as trivial extension algebras.