Modules over the Integral Group Ring of a Non-Abelian Group of Order $pq$

• Author(s): Lee Klingler,
• Publisher: American Mathematical Soc.
• Pages: 125
• ISBN_10: 0821823434
  ISBN_13: 9780821823439
  
• Language: en
• Categories: Mathematics / Algebra / General , Mathematics / Group Theory ,

Description:... If $p,q$ are primes with $p\equiv 1$ $({\rm mod}\,q)$ there is a unique nonabelian group $G$ of order $pq$. The author obtains a classification of the finitely generated $ZG$-modules in sufficient detail to show (1) the Krull-Schmidt theorem fails but the number of indecomposable summands of a decomposable module is independent of the decomposition. He determines (2) the extent to which cancellation of direct summands $(A\oplus B\approx A\oplus C$ implying $B\approx C)$ is valid. He finds (3) the obstruction (in an appropriate semigroup) to the isomorphism of two modules which have isomorphic $t$-adic completions for each prime $t$. The indecomposable modules are constructed by expressing the ring $ZG$ as a pull-back.