Abstract
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.
Highlights
In this paper, we assume that all groups are finite
Question: what can we say about the structure of a group if irreducible character degrees of all proper subgroups are all of prime powers
In order to argue in short, we introduce the following definition
Summary
We assume that all groups are finite. Question: what can we say about the structure of a group if irreducible character degrees of all proper subgroups are all of prime powers. In order to argue in short, we introduce the following definition. Let G be the set of all proper subgroups of a group G. Assume that G is an SDP-group, G is isomorphic to one of the groups: PSL2(q), where q 2rm + 1 ≥ 5 is a prime with r an odd prime or q 2m and 2m − 1 ≥ 3 is a prime, PSL2(9), PSL2(35), S5, or PSL3(3). Journal of Mathematics properties of SDP-groups; in Section 3, we give the structure of almost simple SDP-groups. For a group G, let G be the set of all proper subgroups of G and max G be the set of representatives of maximal subgroups of G. e notation and notions are standard, see [1, 6], for instance
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