A Graph Associated with $|\cd(G)|-1$ Degrees of a Solvable Group

Abstract:

      Let $G$ be a group. We consider the set $\cd(G)\backslash\{m\}$, where $m\in \cd(G)$. We define the graph $\Delta(G-m)$ whose vertex set is $\rho(G-m)$, the set of primes dividing degrees in $\cd(G)\backslash\{m\}$. There is an edge between $p$ and $q$ in $\rho(G-m)$ if $pq$ divides a degree $a\in \cd(G)\backslash\{m\}$. We show that if $G$ is solvable, then $\Delta(G-m)$ has at most two connected components.