少边群的公共非零图
Common non-zero graphs of groups with few edges
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- Amirkabir University of Technology (Tehran Polytechnic)（阿米尔卡比尔理工大学（德黑兰理工学院））
- University of Pretoria（比勒陀利亚大学）
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中文总结 AI 辅助
本文研究有限群公共非零图，证明无边时群为强H_1'-群或Frobenius群（补同构于Q_8），无三角形时可解，单边时同构于S_4或具特定结构，星形图仅当恰有一个非线性不可约特征标。
中文摘要 AI 辅助
设 \\( G \\) 为有限群，考虑其公共非零图，记为 \\( \Gamma_{nv}(G) \\)。在此图中，顶点表示 \\( G \\) 的非线性不可约特征标。若存在一个消失元素 \\( g \in G \\) 使得 \\( \chi_1(g) \chi_2(g) \neq 0 \\)，则连接两个不同顶点 \\( \chi_1 \\) 和 \\( \chi_2 \\) 的边。有限群 \\( G \\) 被称为强 \\( \mathcal{H}_1' \\)-群，如果其每个非线性不可约特征标的消失元素集合等于 \\( G \\) 的消失元素集合。本文证明了文献\cite{ourself}中的猜想1，该猜想指出：若 \\( \Gamma_{nv}(G) \\) 是零图（无边），则 \\( G \\) 要么是强 \\( \mathcal{H}_1' \\)-群，要么是Frobenius群，且其Frobenius补同构于 \\( Q_8 \\)。此外，我们证明若 \\( \Gamma_{nv}(G) \\) 中无三角形，则群可解。我们还研究了有限群 \\( G \\) 的公共非零图仅含一条边的情形。这导致我们得出结论：\\( G \\) 要么同构于 \\( S_4 \\)；要么 \\( \Gamma_{nv}(G) \\) 中的唯一边形如 \\( \{\theta, \zeta\theta\} \\)，其中 \\( \zeta \in \Irr(G/G') \\)，\\( \theta \in \Irr(G|G') \\)，且 \\( G \\) 是某个素数 \\( p \\) 的 \\( p \\)-幂零群，并且对所有 \\( \chi \in \Irr(G|G') - \{\theta, \zeta\theta\} \\)，有 \\( \Van(\chi) = \Van(G) \\)。最后，我们的研究还探讨了公共非零图呈星形结构的群，得出结论：此类结构仅当 \\( G \\) 恰好有一个非线性不可约特征标时才会出现。
英文摘要
Let \( G \) be a finite group, and consider its common non-zero graph, which we denote by \( Γ_{nv}(G) \). In this graph, the vertices represent the non-linear irreducible characters of \( G \). There is an edge connecting two distinct vertices \( χ_1 \) and \( χ_2 \) if there is a vanishing element \( g \in G \) such that \( χ_1(g) χ_2(g) \neq 0 \). A finite group \( G \) is described as a strongly \( \mathcal{H}_1' \)-group if the set of vanishing elements of each of its non-linear irreducible character is equal to the set of vanishing elements of \( G \). In this paper, we prove Conjecture 1 from \cite{ourself}, which states that if \( Γ_{nv}(G) \) is null (has no edge), then \( G \) must either be a strongly \( \mathcal{H}_1' \)-group or a Frobenius group, whose Frobenius complement is isomorphic to \( Q_8 \). Additionally, we show that if there are no triangles in \( Γ_{nv}(G) \), this implies that the group is solvable. We also look at the situation where the common non-zero graph of a finite group \( G \) contains only one edge. This leads us to conclude that \( G \) is either isomorphic to \( S_4 \); or that the single edge in \( Γ_{nv}(G) \) is of the form \( \{θ, ζθ\} \) for some \( ζ\in \Irr(G/G') \) and \( θ\in \Irr(G|G') \), $G$ is a $p$-nilpotent group for some prime $p$ and for all characters \( χ\in \Irr(G|G') - \{θ, ζθ\} \), we find \( \Van(χ) = \Van(G) \). Lastly, our study also explores groups whose common non-zero graph forms a star shape, concluding that such configurations can only occur if \( G \) has exactly one non-linear irreducible character.
