Remarks on Singer cyclic groups and their normalizers.
In this short note the authors consider the structure of a Singer cycle and its normalizer. Let \(\text{GL}(V)\) be the general linear group of dimension \(n\) over a finite field with \(q\) elements. Then a Singer cycle is a cycle which acts as a regular subgroup on the non-zero elements of \(V\) and so is of order \(q^n-1\). They are mainly concerned with the case when \(n=3\). Here they define \(\sigma\) to be the Frobenius automorphism induced on the field, \(F\), of order \(q^3\). Identifying \(F\) with the vector space \(V\) with a normal basis they completely describe both the normalizer and centralizer of \(\sigma\). Interestingly the structure of the centralizer depends on the congruence of \(q\) modulo \(3\). Two simple comments, the inequality in the first line of the proof of Proposition 2.4 is the wrong way round, though correct in the next line. Also after the statement that `from now on we assume \(n=3\)' it would be easier to recall that if a slight change of notation was used.
scientific article; zbMATH DE number 2091507