Determining group structure from the sets of character degrees. (Q961029)

The purpose of the paper under review is to study the structure of a finite solvable group with a restricted set of character degrees. This is a contribution to the general question which sets of numbers can occur as sets of character degrees of a group and how this set influences the group structure. The results are a continuation of a study begun by \textit{M. L. Lewis} [in J. Algebra 206, No. 1, 235-260 (1998; Zbl 0915.20003); Corrigendum, ibid. 322, No. 6, 2251-2252 (2009; Zbl 1173.20303)]. To give a flavor of the topic, we state the first of the two main results of the paper in detail. First, for a prime \(p\) and an integer \(m>1\) we say that the ordered pair \((p,m)\) is `strongly coprime' if \(p\) does not divide \(m\), and also \(p\) does not divide \(u-1\) for all prime powers \(u\) dividing \(m\) with \(1<u<m\). Theorem A: Let \(p\) be a prime and \(m,n\) coprime integers greater than 1 such that \((p,m)\) and \((p,n)\) are strongly coprime. Let \(G\) be a finite solvable group with \(\text{cd}(G)=\{1,p,n,m,pn,pm\}\). Then \(\text{cd}(\mathbf O^p(G))=\{1,m,n\}\) and one of the following holds: (1) \(G=A\times B\), where \(\text{cd}(A)=\{1,p\}\) and \(\text{cd}(B)=\{1,n,m\}\); (2) there is a prime \(t\) such that \(G\) has a normal Sylow \(t\)-subgroup \(T\) with \(\text{cd}(T)=\{1,t^l\}\) for some \(l\geq 2\), \(t^l\in\{n,m\}\), \(\mathbf O^p(G)/T\) is Abelian, the Fitting height of \(\mathbf O^p(G)\) is 2, and so the Fitting height of \(G\) is at most 3. -- There do exist examples showing that (2) really does occur.