Semi-rational solvable groups
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David Chillag
Abstract
An element x of a finite group G is called rational if all generators of the group 〈x〉 are contained in a single conjugacy class. If all elements of G are rational, then G itself is called rational. It was proved by Gow that if G is a rational solvable group then π(G) ⊂ {2, 3, 5}. We call x ∈ G semi-rational if all generators of 〈x〉 are contained in a union of two conjugacy classes. Furthermore, we call x ∈ G inverse semi-rational if every generator of 〈x〉 is conjugate to either x or x–1. Then G is called semi-rational (resp. inverse semi-rational) if all elements of G are semi-rational (resp. inverse semi-rational). We show that if G is semi-rational and solvable then π(G) ⊂ {2, 3, 5, 7, 13, 17}, and if G is inverse semi-rational and solvable then 17 ∉ π(G). If G has odd order, then it is semi-rational if and only if it is inverse semi-rational. In this case we describe the structure of G.
© de Gruyter 2010
Articles in the same Issue
- A note on the Green theory in the Clifford-theoretic context of a defect zero p-block of a normal subgroup of a finite group
- A note on a conjecture of K. Harada and strongly p-embedded Frobenius subgroups
- Carter–Payne homomorphisms and branching rules for endomorphism rings of Specht modules
- Some linear actions of finite groups with q′-orbits
- Semi-rational solvable groups
- Finite non-abelian 2-groups such that any two distinct minimal non-abelian subgroups have cyclic intersection
- A note on the p-supersolvability of finite groups
- Vertex-transitive tournaments of order a product of two distinct primes
- On the derived length of the unit group of a group algebra
- On the subgroups with non-trivial Möbius number
- Subgroups of free groups and primitive elements
- Small index subgroups of the mapping class group
Articles in the same Issue
- A note on the Green theory in the Clifford-theoretic context of a defect zero p-block of a normal subgroup of a finite group
- A note on a conjecture of K. Harada and strongly p-embedded Frobenius subgroups
- Carter–Payne homomorphisms and branching rules for endomorphism rings of Specht modules
- Some linear actions of finite groups with q′-orbits
- Semi-rational solvable groups
- Finite non-abelian 2-groups such that any two distinct minimal non-abelian subgroups have cyclic intersection
- A note on the p-supersolvability of finite groups
- Vertex-transitive tournaments of order a product of two distinct primes
- On the derived length of the unit group of a group algebra
- On the subgroups with non-trivial Möbius number
- Subgroups of free groups and primitive elements
- Small index subgroups of the mapping class group
