Intersections of Matrix Algebras and Permutation Representations of PSL(n, q)

✍ Scribed by J. Siemons; A. Zalesskii

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 288 KB
• Volume: 226
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8201

✦ Synopsis

If G is a group, H a subgroup of G, and ⍀ a transitive G-set we ask under what Ž < <. conditions one can guarantee that H has a regular orbit s of size H on ⍀.

Ž . Ž . Here we prove that if PSL n, q : G : PGL n, q and H is cyclic then H has a Ž . regular orbit in every non-trivial G-set with few exceptions . This result is obtained via a mixture of group theoretical and ring theoretical methods: Let R be the ring of all n = n matrices over the finite field F and let Z be the subring of scalar matrices. We show that if A and M are proper subrings of R containing Z, Ž . and if A is commutative and semisimple, then there exists an element x g SL n, F y1 < < such that xAx l M s Z or n s 2 s F .