We conjecture that if G is a finite primitive group and if g is an element of G, then either the element g has a cycle of length equal to its order, or for some r, m and k, the group G≤. Sym( m)wrSym( r), preserving a product structure of r direct copies of the natural action of Sym( m) or Alt( m) on k-sets. In this paper we reduce this conjecture to the case that G is an almost simple group with socle a classical group.
Giudici, M., Praeger, C., & Spiga, P. (2015). Finite primitive permutation groups and regular cycles of their elements. JOURNAL OF ALGEBRA, 421(SI), 27-55 [10.1016/j.jalgebra.2014.08.015].
Finite primitive permutation groups and regular cycles of their elements
SPIGA, PABLOUltimo
2015
Abstract
We conjecture that if G is a finite primitive group and if g is an element of G, then either the element g has a cycle of length equal to its order, or for some r, m and k, the group G≤. Sym( m)wrSym( r), preserving a product structure of r direct copies of the natural action of Sym( m) or Alt( m) on k-sets. In this paper we reduce this conjecture to the case that G is an almost simple group with socle a classical group.
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