Conjugacy in groups with dihedral 3-normalisers

Normalisers in Limit Groups

Let Γ be a limit group, S ⊂ Γ a non-trivial subgroup, and N the normaliser of S. If H1(S,Q) has finite Q-dimension, then S is finitely generated and either N/S is finite or N is abelian. This result has applications to the study of subdirect products of limit groups. Limit groups (or ω-residually free groups) have received a good deal of attention in recent years, largely due to the work of Z. ...

Constructing Normalisers in Finite Soluble Groups

This paper describes algorithms for constructing a Hall n-subgroup H of a finite soluble group G and the normaliser No(H). If G has composition length n, then H and No(H ) can be constructed using O(n ~ log IGI) and O(n ~ log IGI) group multiplications, respectively. These algorithms may be used to construct other important subgroups such as Carter subgroups, system normalisers and relative sys...

Sumsets in dihedral groups

Let Dn be the dihedral group of order 2n. For all integers r, s such that 1 ≤ r, s ≤ 2n, we give an explicit upper bound for the minimal size μDn (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of Dn of cardinality r and s respectively. It is shown by construction that μDn (r, s) is bounded above by the known value of μG (r, s), where G is any abelian ...

Polynomials Associated with Dihedral Groups

Abstract. There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The ...

Dihedral Groups