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Idempotent ideals in integral group rings
Lachman, Dominik ; Příhoda, Pavel (advisor) ; Šaroch, Jan (referee)
This thesis concerns following hyphotesis: whenever I is two-sided idem- potent ideal in group ring ZSn, such that IQ is non-trivial ideal of QSn, IQ has to be so called augmentation ideal. The vylidity of this hypothesis would give us weak version of the fact that in the case of solvable group G, there are no two-sided non-trivial idempotent ideals in ZG. At first I desctibe methodt how to calculate idempotent ideals in ZSn and then show that hy- pothesis holds in the case of S5, but fail in the case of ZS5. In theoretic part, I firstly switch to local point of view and describe two-sided idempo- tent ideals in Z(p)Sn, for primes p dividing order of group Sn, as trace ide- als of finitely generated projective Z(p)Sn-modules. Next, I describe functor −⊗Z(p) Q : Proj(Z(p)Sn) → Mod(QSn) using the language of Grothendiecks groups by matrix E. Matrix E shows to be transposition of decomposition matrix, which we can calculate using Braeur's character. 1
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