Subgroups of groups of central type

Abstract:

Let $\lambda$ be a linear character on the center Z of a finite group Z of a finite group H, such that (1) ${\lambda ^H} = \sum \nolimits _{i = 1}^p {{\phi _i}(1){\phi _i}}$ where the ${\phi _i}$’s are inequivalent irreducible characters on H of the same degree, and (2) if $\sum \nolimits _{i = 1}^p {{m_i}{\phi _i}(x) = 0}$ for some $x \in H$ and nonnegative integers ${m_i}$, then either ${\phi _i}(x) = 0$ for all i or ${m_i} = {m_j}$ for all i, j. The object of the paper is to describe finite groups which satisfy conditions (1) and (2) in terms of the multiplication of the group. If S is a p Sylow subgroup of the group H, and $R = S \cdot Z$, then H satisfies conditions (1) and (2) if and only if (a) $\{ x \in H:{x^{ - 1}}{h^{ - 1}}xh \in Z \Rightarrow \lambda ({x^{ - 1}}{h^{ - 1}}xh) = 1,h \in H\} /Z$ consists of elements of order a power of p in $H/Z$, and these elements form p conjugacy classes of $H/Z$, and (b) the elements of $\{ x \in R:{x^{ - 1}}{r^{ - 1}}xr \in Z \Rightarrow \lambda ({x^{ - 1}}{r^{ - 1}}xr) = 1,r \in R\} /Z$ form p conjugacy classes of $R/Z$.