Representations and primitive central idempotents of a finite solvable group (original) (raw)
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ON A CLASS OF GROUP ALGEBRAS AND THEIR PRIMITIVE IDEMPOTENTS
Let F denotes a finite field having q = t (t is a prime,s ≥ 1 ) elements .We Consider the group algebra FqG where G is a finite group of order n. When F contains a primitive nth root of unity (implying that characteristic of Fqdoes not divide n) there is a known formula for obtaining the fall system of primitive idempotent of FqG which is also an orthogonal Fqbasis for the group algebra FqG. The approach is via group character which are homeomorphisms from G into the cyclic group Fq* of non zero elements of Fq.Earlier, ManjuPruthi ,S.K .Aroraetal have obtained expressions for primitive idempotent of the group algebra FqCm where Cm is a cyclic group of order m . They used techniques from the notation of q-cyclotomiccosets modulo m. The method presented here explains the role of “semi simplicity”of the ring Fq [x]∕(x m -1) and its connection with the group algebra FqCm. Inparticular examples are given when the group G is cyclic of order n or the dihedral group Dn of order 2n . KEYWORDSCharacter group, group algebra, semi
ON A CLASS OF GROUP ALGEBRAS AND THEIR PRIMITIVE IDEMPOTENTS 1
Let F denotes a finite field having q = t (t is a prime,s ≥ 1) elements .We Consider the group algebra F q G where G is a finite group of order n. When F contains a primitive n th root of unity (implying that characteristic of F q does not divide n) there is a known formula for obtaining the fall system of primitive idempotent of F q G which is also an orthogonal F q basis for the group algebra F q G. The approach is via group character which are homeomorphisms from G into the cyclic group F q * of non zero elements of F q .Earlier, Manju Pruthi ,S.K .Arora etal have obtained expressions for primitive idempotent of the group algebra F q C m where C m is a cyclic group of order m. They used techniques from the notation of q-cyclotomic cosets modulo m. The method presented here explains the role of " semi simplicity " of the ring F q [x] (x m-1) and its connection with the group algebra F q C m. Inparticular examples are given when the group G is cyclic of order n or the dihedral group D n of order 2n .
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On Monomial Characters and Central Idempotents of Rational Group Algebras
Communications in Algebra, 2004
We give a method to obtain the primitive central idempotent of the rational group algebra QG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-bysupersolvable groups this method takes a particularly easy form that can be used to compute the Wedderburn decomposition of QG.