On imprimitive multiplicity-free permutation groups the degree of which is the product of two distinct primes. (English) Zbl 1171.20002
Let \(\mathcal{PQ}\) denote the set of positive integers \(n\) such that \(n\) is a product of two distinct primes such that \(\gcd(n,\varphi(n))=1\); let \(\mathcal R\) be the set of all \(n\in\mathcal{PQ}\) such that any imprimitive permutation group of degree \(n\) is multiplicity-free.
The main result of the paper is Theorem 1.1 which says that \(\mathcal{PQ}\setminus\mathcal R=\{pq\mid q\equiv 2\pmod p\), where \(q\) is a Fermat prime and \(p\) is a prime}.
The result depends on the classification of permutation groups of prime degree (via the classification of finite simple groups).
The main result of the paper is Theorem 1.1 which says that \(\mathcal{PQ}\setminus\mathcal R=\{pq\mid q\equiv 2\pmod p\), where \(q\) is a Fermat prime and \(p\) is a prime}.
The result depends on the classification of permutation groups of prime degree (via the classification of finite simple groups).
Reviewer: Wolfgang D. Knapp (Tübingen)
MSC:
| 20B10 | Characterization theorems for permutation groups |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
Keywords:
finite permutation groups; imprimitive permutation groups; permutation character; multiplicity-free characters; centralizer algebras; Fermat primesReferences:
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