Abstract
It is established that degree 2|A|+1 of irreducible complex linear group with the group A of cosimple automorphisms of odd order is a prime number and proved that if degree 2|H|+1 of π-solvable irreducible complex linear group G with a π-Hall TI-subgroup H is not a prime power, then H is Abelian and normal in G.
Highlights
Suppose that G is a finite group and A is such a group of its nontrivial automorphisms that |G|, |A| 1
A is called a group of cosimple automorphisms of the group G and the semidirect product Γ GA of the group G and the group A is a group
The theorem, proved in the series of papers 1–3, implies that if G is a finite irreducible complex linear group of degree n < 2|A| with a nontrivial strong-centralized odd-order group A of cosimple automorphisms, n |A| − 1, |A| 1, 2 |A| − 1 or 2|A| − 1 and n is a degree of a certain prime number
Summary
It is established that degree 2|A| 1 of irreducible complex linear group with the group A of cosimple automorphisms of odd order is a prime number and proved that if degree 2|H| 1 of π-solvable irreducible complex linear group G with a π-Hall T I-subgroup H is not a prime power, H is Abelian and normal in G
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