Determining group structure from sets of irreducible character degrees
The principal results of this paper are the following: Theorems A + B. Let \(p,q,r\), and \(s\) be distinct primes. (A) If \(G\) is a group such that \(\text{cd}(G)=\{1,p,q,r,pq,pr\}\), then \(G=A\times B\), where \(\text{cd}(A)=\{1,p\}\) and \(\text{cd}(B)=\{1,q,r\}\). (B) If \(G\) is a group such that \(\text{cd}(G)=\{1,p,q,r,s,pr,ps,qr,qs\}\), then \(G=A\times B\), where \(\text{cd}(A)=\{1,p,q\}\) and \(\text{cd}(B)=\{1,r,s\}\). Isaacs and Passman have classified groups all of whose nonlinear irreducible characters have prime degrees. Therefore, the structure of \(A\), \(B\) of Theorems A and B is known. The paper contains intermediate results of independent interest. We will present some of them. Lemma 3.2. Let \(S\) act coprimely on \(G\). If \([G,S]\leq Z(G)\), then \(G=[G,S]\times C_G(S)\). Lemma 3.2 resembles Fitting's Lemma. In Sec. 4 the groups \(G\) with \(\text{cd}(G)=\{1,m,n\}\) with coprime \(m,n\) are considered. Theorem 5.2. Let \(p,q\), and \(r\) be distinct primes and write \(K=O^p(G)\). Assume that \(\text{cd}(K)=\{1,q,r\}\subseteq\text{cd}(G)\). Then \(G\) splits over \(K\) and there is a complement \(N\) such that the following hold. (1) \([K,N]\leq Z(K)\) and \(Z(K)\) is a \(p'\)-group. (2) \(G=C_K(N)\times N[K,N]\). (3) \(\text{cd}(K)=\text{cd}(C_G(N))\). Retaining the notation and hypotheses of Theorem 5.2, one proves that if \(\text{cd}(G)\) is as in Theorem A, then the conclusion of Theorem A holds. Theorems 7.1 + 7.2. Let \(G\) be a solvable group of Theorem A. (7.1) If \(\text{cd}(G /K)=\{1,p\}\) for \(K\triangleleft G\), then \(\text{cd}(O^p(G))=\{1,q,r\}\). (7.2) If \(K\) is a normal subgroup of \(G\) such that \(G/K\) is a Frobenius group with cyclic complement of order \(pq\) or \(pr\) and elementary abelian \(s\)-group kernel \(N/K\), where \(s\) is some prime, then \(\text{cd}(O^p(G))=\{1,q,r\}\). The author also presents an example to show that, given any coprime \(m,n\in\mathbb{N}-\{1\}\), there exists a group \(G\) with \(\text{cd}(G)=\{1,m,n,mn\}\), where \(G\) is not a nontrivial direct product.
scientific article; zbMATH DE number 1214151