On Groups Whose Irreducible Character Degrees of All Proper Subgroups are All Prime Powers

Isaacs, Passman, and Manz have determined the structure of finite groups whose each degree of the irreducible characters is a prime power. In particular, if G is a nonsolvable group and every character degree of a group G is a prime power, then G is isomorphic to S × A , where S ∈ A 5 , PSL 2 8 and A is abelian. In this paper, we change the condition, each character degree of a group G is a prime power, into the condition, each character degree of the proper subgroups of a group is a prime power, and give the structure of almost simple groups whose character degrees of all proper subgroups are all prime powers.

Nonsolvable groups with four character codegrees

Let [Formula: see text] be a finite group and the codegree of an irreducible character [Formula: see text] is the number [Formula: see text]. In this paper, we consider nonsolvable groups with few character codegrees and prove that if a nonsolvable group [Formula: see text] has exactly four character codegrees, then [Formula: see text], where [Formula: see text].

On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups

LetGbe a finite group and𝒩𝒞(G)the set of the numbers of conjugates of noncyclic proper subgroups ofG. We prove that (1) if|𝒩𝒞(G)|≤2, thenGis solvable, and (2)Gis a nonsolvable group with|𝒩𝒞(G)|=3if and only ifG≅PSL(2,5)orPSL(2,13)orSL(2,5)orSL(2,13).

GENERALIZING A THEOREM OF HUPPERT AND MANZ

Huppert and Manz proved that if G is a nonsolvable group where all the character degrees are square-free, then G ≅ A7 × S, where S is a solvable group with no degree divisible by 2, 3, 5, or 7. We weaken the hypothesis of Huppert and Manz's theorem to prove the following generalization. If G is a nonsolvable group and 4 divides no character degree of G, then G ≅ A7 × S, where S is a solvable group with no degree divisible by 2.

On collineation groups of translation planes of orderq4

LetPbe an affine translation plane of orderq4admitting a nonsolvable groupGin its translation complement. IfGfixes more thanq+1slopes, the structure ofGis determined. In particular, ifGis simple thenqis even andG=L2(2s)for some integersat least2.

SL(2,5) and Frobenius Galois Groups Over Q

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we ObtainTHEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.