#TITLE_ALTERNATIVE# - PURWA SANTIKA (NIM: 30110003), ADITYA
Substructure of centre of a group algebra called p-regular subspace is related to conjugacy class of group elements whose order is not divisible by p > 0 a prime. This subspace is spanned by class sums of p-regular conjugacy classes, i.e. conjugacy classes of elements whose order is not divisible by p. The dimension of this subspace is related to the number of isomorphism classes of simple module over the group algebra. The p-reguler subspace is not always closed under multiplication. Meyer gave examples of groups whose p-regular subspace is closed and not closed under multiplication. Derived category is a triangulated category constructed from an existing category. This category is defined in such a way that a quasi-isomorphism, a morphism that induces isomorphism between the homologies, is an isomorphism in derived category. Brou´e proved that equivalence between two derived categories induces perfect isometry between them. Fan and K¨ulshammer showed that p-regular subspace is invariant under perfect isometry. As a generalization of that, we investigate wether p-regular subspace invariant or not under derived equivalence. This dissertation identifies p-regular subspace as the dual of the intersection of the images of pn-power maps in commutator quotient space, for every positive integer n. We use this identification to study the invariance of p-regular subspace under derived equivalence. As this identification does not depend on the order of elements in group, we can extend this definition to any symmetric algebra. As examples, we determine p-regular subspaces of Nakayama algebra, algebra of dihedral type and algebra of semidihedral type. Furthermore, we also study the invariance of p-regular subspace of symmetric path algebras under derived equivalence.