Let \(G\) be a finite group and let \(A\) be a group of automorphisms of \(G\) such that \((|G|,|A|)= 1\). Then \(A\) is called a group of cosimple automorphisms of \(G\) and the semidirect product \(\Gamma= GA\) of \(G\) and \(A\) is a group. If \(C_{G}(a)= C_{G}(A)\) for each element \(a\in A^{\#}\) then \(A\) is said to be a strong-centralized group of cosimple automorphisms of \(G\).
Condition B. We say that the group \(\Gamma =GA\) satisfies Condition B if \(G\triangleleft \Gamma\), \((|A|,|G|)=1\), \(A\) is an odd-order group that is not normal in \(\Gamma\), \(C_{G}(a)= C_{G}(A)= C\) for each element \(a\in A^{\#}\), and \(G\) has a faithful irreducible complex character of degree \(n\) which is \(a\)-invariant for at least one element \(a\in A^{\#}\).
From the theorem proved in [1–3], it is obvious that if \(n<2|A|\) and A is of odd order then \(n={\left|A\right| } -1,\)
\({\left|A\right| }+1,\) \(2({\left|A\right| }-1)\), or
\(2{\left|A\right| }-1\), and \(n\) is a degree of a certain prime number. The paper [4] states that if the group \(\Gamma\) satisfies Condition B and \(n=2|A|+1\), then \(n\) is also a prime power. Hence, \(n\) is divisible by the degree \(f\) of a prime such that \(f\equiv -1\) or \(1(\)mod\(|A|)\). The paper [3] hypothesizes the fairness of this statement for an arbitrary number \(n\).
Suppose that \({\left|A\right| } =p\) is a prime number. Then, from the above-mentioned theorem, we obtain Isaacs's result [5] for the groups that have the above-named property, and the appropriate result obtained by Newton [6] follows from the hypothesis.
Theorem.
Assume the group \(\Gamma\) satisfies Condition B. Then \(n\) will divide by the degree \(f\) of a certain prime number such that \(f\equiv -1\) or \(1(\)mod\(|A|)\).
References
A. A. Yadchenko, On \(\Pi\)-solvable irreducible linear groups with a Hall \(TI\)-subgroup of odd order. pt. I, Works of the Institute of Mathematics of NASB, 16, N2 (2008), 118–130.
A. A. Yadchenko, On \(\Pi\)-solvable irreducible linear groups with a Hall \(TI\)-subgroup of odd order. pt. II, Works of the Institute of Mathematics of NASB, 17, N2 (2009), 94–104.
A. A. Yadchenko, On \(\Pi\)-solvable irreducible linear groups with a Hall \(TI\)-subgroup of odd order. pt. III, Works of the Institute of Mathematics of NASB, 18, N2 (2010), 99–114.
A. A. Yadchenko, On Irreducible Linear Groups of Nonprimary Degree, ISRN Algebra, 2011 (2011), Article ID 868096, doi: 10.5402/2011/868096, 20 p.
I. M. Isaacs, Complex p-solvable linear groups, J. Algebra, 24, N3 (1973), 513–530.
B. Newton, On the degrees of complex p-solvable linear groups, J. Algebra, 288 (2005), 384–391.
