15.2 (2002)

By a theorem of W. Burnside, if $\chi \in \text{Irr}(G)$ and $\chi(1) > 1$, then there exists $x \in G$ such that $\chi(x) = 0$, that is, only the linear characters are “nonvanishing”. It is interesting to consider the dual notion of nonvanishing elements of a finite group $G$, that is, the elements $x \in G$ such that $\chi(x) \neq 0$ for all $\chi \in \text{Irr}(G)$.
$\qquad$ a) It is proved (I. M. Isaacs, G. Navarro, T. R. Wolf, J. Algebra, 222, no. 2 (1999), 413–423) that if $G$ is solvable and $x \in G$ is a nonvanishing element of odd order, then $x$ belongs to the Fitting subgroup $\mathfrak{F}(G)$. Is this true for elements of even order too? (M. Miyamoto showed in 2008 that every nontrivial abelian normal subgroup of a finite group contains a nonvanishing element.)
$\qquad$ b) Which nonabelian simple groups have nonidentity nonvanishing elements? (For example, $A_7$ has.)