2006/10/31 by Ben Webster
Mathematics · #Advanced Topics in Algebra #Cardinality (data modeling) #Combinatorics #Commutative property #Commutative ring #Discrete mathematics #Field (mathematics) #Finite Group Theory Research #Finite field #Group (periodic table) #Mathematics #Permutation (music) #Permutation group #Pure mathematics #Ring (chemistry) #Rings, Modules, and Algebras #math.GR #msc:20C99
paper · pdf · doi:10.1016/j.jalgebra.2007.07.007
published as J. Algebra 317 (2007), no. 1, 306-323 · v2: fixed proof of Lemma 2.1
openalex publication_date 2007/08/10 · arxiv created 2008/07/27 · arxiv updated 2010/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Two G-sets (G a finite group) are called linearly equivalent over a commutative ring k if the permutation representations k[X] and k[Y] are isomorphic as modules over the group algebra kG. Pairs of linearly equivalent non-isomorphic G-sets have applications in number theory and geometry. We characterize the groups G for which such pairs exist for any field, and give a simple construction of these pairs. If k is \Q, these are precisely the non-cyclic groups. For any non-cyclic group, we prove that there exist G-sets which are non-isomorphic and \lineq over \Q, of cardinality ≤ 3(#G)/2. Also, we investigate a construction of P. Beaulieu which allows us to construct pairs of transitive linearly equivalent Sn-sets from arbitrary G-sets for an arbitrary group G. We show that this construction works over all fields and use it construct, for each finite set \mc P of primes, Sn-sets linearly equivalent over a field k if and only if the characteristic of k lies in \mc P.