On the number of characters in blocks of finite general linear, unitary and symmetric groups (Q790239)

The author proves the following conjecture of R. Brauer to be true for the classes of groups mentioned in the title: If G is a finite group, r a prime number, B an r-block of G, k(B) the number of ordinary irreducible characters in B, d(B) the defect and \(\Delta\) (B) a defect group of B, then \(k(B)\leq r^{d(B)}=| \Delta(B)|,\) if G is a finite symmetric group or \(2\neq r\nmid q\) and \(G=GL(n,q)\) or \(G=U(n,q).\) Furthermore he shows that in these cases we have \(k_ 0(B)\leq | \Delta(B)/\Delta(B)'|,\) where now \(k_ 0(B)\) denotes the number of ordinary irreducible characters of height 0 in B, and \(\Delta\) (B)' is the commutator subgroup of \(\Delta\) (B).