| A Graph Associated with $|\cd(G)|-1$ Degrees of a Solvable Group |
| Received:June 11, 2009 Revised:September 15, 2009 |
| Key Words:
solvable groups irreducible character degrees.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10871032), Innovation Project for the Development of Science and Technology (IHLB) (Grant No.201098) and the Specific Research Fund of the Doctoral Program of Higher Education of China (Grant No.20060285002). |
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| 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. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2011.01.021 |
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