Let $G$ be a finite group. Its length $\ell(G)$ is the length $n$ of any Jordan-Hölder filtration $1=G_n\subset\cdots\subset G_0=G$ ($G_i$ is normal in $G_{i-1}$ and $G_{i-1}/G_i$ is simple).

Can $\ell(G)$ be read (easily) in the table of characters of $G$ ?

For instance $\ell(G)=1$ (that is $G\ne1$ is simple) if and only if there does not exist a character $\chi\ne\bf1$ such that $\chi(g)=\chi(e)$ for some $g\ne e$.

Let $G$ be a finite group. Its length $\ell(G)$ is the length $n$ of any Jordan-Hölder filtration $1=G_n\subset\cdots\subset G_0=G$ ($G_i$ is normal in $G_{i-1}$ and $G_{i-1}/G_i$ is simple).

Can $\ell(G)$ be read (easily) in the table of characters of $G$ ?

For instance $\ell(G)=1$ (that is $G\ne1$ is simple) if and only if there does not exist a character $\chi\ne\bf1$ such that $\chi(g)=\chi(e)$ for some $g\ne e$.

A suggestion for starting: consider the case of a $p$-group of order $p^m$. Because its J-H factors are $p$-groups and simple, they are all isomorphic to $C_p$ and therefore $\ell(G)=m$. Is there any general fact concerning the table of $p$-groups, that is directly related to $m$ ?

Let $G$ be a finite group. Its length $\ell(G)$ is the length $n$ of any Jordan-Hölder filtration $1=G_n\subset\cdots\subset G_0=G$ ($G_i$ is normal in $G_{i-1}$ and $G_{i-1}/G_i$ is simple).

Can $\ell(G)$ be read (easily) in the table of characters of $G$ ?

For instance $\ell(G)=1$ (that is $G\ne1$ is simple) if and only if there exists a character $\chi\ne\bf1$ such that $\chi(g)=\chi(e)$ for some $g\ne e$.

Let $G$ be a finite group. Its length $\ell(G)$ is the length $n$ of any Jordan-Hölder filtration $1=G_n\subset\cdots\subset G_0=G$ ($G_i$ is normal in $G_{i-1}$ and $G_{i-1}/G_i$ is simple).

Can $\ell(G)$ be read (easily) in the table of characters of $G$ ?

For instance $\ell(G)=1$ (that is $G\ne1$ is simple) if and only if there does not exist a character $\chi\ne\bf1$ such that $\chi(g)=\chi(e)$ for some $g\ne e$.