Squares of characters that are the sum of all irreducible characters (Q1280728)

Consider the groups satisfying the following condition: (*) Let \(G\) be a nonidentity finite group for which there exists an irreducible character \(\chi\) such that \(\chi^2=\sum_{\psi\in\text{Irr}(G)}\psi\). It follows from (*) that \(\chi\) is faithful, real-valued (Theorem 2.4) and the center of \(G\) must be trivial. Moreover, \(\chi\) is afforded by a real representation and every real-valued irreducible character of \(G\) has Frobenius-Schur indicator equal to \(+1\). It follows that \(\chi(1)^2=1+t+\sum\chi(1)\), where \(t\) is the number of involutions in \(G\), and the sum extends over all irreducible characters of \(G\) that are not real-valued. (The last sum is even.) Since \(1+t\) is the number of solutions of \(x^2=1\) in \(G\), it is even, unless \(G\) is of odd order. In the last case \(G\) is solvable so it is does not satisfy (*), as the theorem below shows. Thus, we see that, in the case under consideration, \(\chi(1)\) is even. The authors conjectured that \(\chi(1)\) is a power of 2 always. As the following theorem shows this is the case if \(G\) is solvable. Theorems 1.2. Assume \((G,\chi)\) satisfies (*). If \(G\) is a tower of groups of prime orders and \(\text{PSL}_2(2^n)\), \(n>1\), then \(G\) is a direct product of groups that are isomorphic to \(\text{PSL}(2^k)\), \(k\geq 1\). Note that if \(\chi\) is nonlinear irreducible then \(\chi^3\) is not multiplicity free, so it is not a sum of pairwise distinct irreducible characters of \(G\) (Lemma 2.1). It is clear that the class of (*) is fairly small. In fact, for (*) groups \(G\), the number \(s(G)=\sum_{\theta\in\text{Irr}(G)}\theta(1)\) is a square and all prime divisors of \(s(G)\) divide \(| G|\). This allows us to eliminate all sporadic groups. Note that if \(G=\text{PSL}_2(q)\), where \(q\) is a power of \(2\), satisfies \(s(G)=q^2\) and it, as we know, is a (*)-group. Let \(G\) be a (*)-group. Then all sporadic groups, the groups \(\text{PSL}_2(q)\) where \(q>5\) is a power of an odd prime, and the group \({^2F_4}(2)'\) are not epimorphic images of \(G\).