Primitive Decompositions Of Idempotents Of Group Algebras Of Dihedral Groups And Generalized Quaternion Groups

The Wedderburn theorem indicates that every semisimple algebra is unique-ly expressible as a direct sum of matrix algebras;conversely,every direct sum of matrix algebras are semi simple.In this paper,the primitive decompositions of idempotents of the dihedral group algebras （?）[D_2n]and generalized quaternion group algebras （?）[Q_4m]are calculated by their Wedderburn decomposition.In-spired by the orthogonality relations of the character tables of these two families of groups,we obtain two sets of trigonometric identities.Furthermore,a group algebra isomorphism between （?）[D₈]and （?）[Q₈] is described,under which our t-wo complete sets of primitive orthogonal idempotents of these two group algebras correspond to each other bijectively.Furthermore,we study the linear preserving problem of matrix algebra M_n（（?））.By using representation theory,we characterize the automorphisms of M_n（（?）） that keep the principal diagonal elements unchanged,further determine that there exists a one-to-one correspondence between the com-plete set of orthogonal primal idempotent elements of matrix algebra M_n（（?）） and the homogeneous space GL_n（（?））/Diag_n（（?））.