Finite groups whose all proper subgroups have only irreducible characters of square-free degrees (Q6610144)

Let \(G\) be a finite group, Irr\((G)\) the set of complex irreducible characters of \(G\), and cd\((G)\) the set of degrees of characters lying in \(G\), i.e. cd\((G) = \{\chi (1)\colon \chi \in \) Irr\((G)\}\). Denote by \(\Sigma (G)\) the set of subgroups of \(G\), and by \(\Sigma_(G)\) the set of proper subgroups of \(G\). We say that \(G\) is a PD-group if \(\chi (1)\) is square-free, for each \(H \in \Sigma (G)\) and any \(\chi \in \)Irr\((H)\); \(G\) is said to be a non-PD-group, otherwise. The group \(G\) is called an SPD-group if the set \(\Sigma_p(G)\) consists of PD-groups; \(G\) is called a non-PD-group, otherwise. The group \(G\) is called an SPD-group if the set \(\Sigma_p(G)\) consists of PD-groups; \(G\) is called a non-SPD-group, otherwise. Abelian finite groups are always considered to be PD-groups and SPD-groups. It is known that there exist finite groups whose degrees are square-free, which are neither PD nor SPD. An example is provided by the alternating group \(A_7\) of degree \(7\); as noted by the authors of the present paper, cd\((A_7) = \{1, 6, 10, 14, 15, 21, 35\}\) (which is easy to see from page 10 of [Conway, J. H.; Curtis, R.T.; Norton, S.P.; Parker, R.A.; Wilson, R.A. [Thackray, J.G.]: Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With comput. assist. from J. G. Thackray. Oxford: Clarendon Press. XXXIII, 252 p. (1985; Zbl 0568.20001). In addition, it is known (see Huppert, B.; Manz, O.: Arch. Math. 45, 125-132 (1985; Zbl 0574.20006) that if \(G\) is a group such that cd\((G)\) is a set of square-free numbers, then \(G\) is either solvable or decomposable into the direct product \(A_7 \times N\), where \(N\) is a solvable group; in the former case, the derived length and the nilpotent length of \(G\) are at most equal to \(4\) and \(3\), respectively. The paper under review considers the structure of PD-groups and of SPD-groups. Its first main result states that finite PD-groups are solvable. In the rest of the paper, the authors focus their attention on non-solvable finite SPD-groups. Assuming that \(G\) is such a group, they show that it is isomorphic to one of the following groups: (i) the projective special linear group PSL\(\sb 2(q)\), for certain \(q \in \{2^p, 3^p, r\}\) such that \(p, r\) are primes and \((q - 1)/\gcd (2, q - 1)\) is square-free; (ii) the special linear group SL\(\sb 2(3^p)\) with \(3^p - 1\) square-free; (iii) SL\(\sb 2(q)\) with \(q\) an odd prime congruent to \(3\) or \(27\) modulo \(40\), such that \(q - 1\) is square-free. Also, the paper contains information about the isomorphism classes of \(G\) in case it is a non-abelian simple SPD-group. The proof of both results rely on Thompson's classiffication of non-abelian finite simple groups whose proper subgroups are solvable (see Corollary~1 in: [Thompson, J.G., Bull Am. Math. Soc. 74, 383-437 (1968; Zbl 0159.30804)]. Finally, it is proved that a finite SPD-group is solvable, provided that it is without a section isomorphic to PSL\(\sb 2(q)\).