有限p-群的表示维数
On the representation dimension of finite $p$-groups
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中文总结 AI 辅助
该研究确定了有限p-群的表示维数的加性性质，计算了几类非阿贝尔p-群及所有阶为p⁶（p≥5）群的表示维数。
中文摘要 AI 辅助
对于有限群G，其表示维数δ(G)是G的忠实复表示的最小维数。我们证明：当H和K为固定素数p的p-群时，δ(H×K)=δ(H)+δ(K)，并举例说明该加性对幂零群而言可能不成立。我们还确定了几类重要非阿贝尔p-群（即VZ群、Camina群、亚循环群）的δ(G)，并按同倾类计算了所有阶为p⁶（p≥5）的群的δ(G)。
英文摘要
For a finite group $G$, the representation dimension $δ(G)$ is the least dimension of a faithful complex representation of $G$. We prove that $δ(H\times K) = δ(H) + δ(K)$ whenever $H$ and $K$ are $p$-groups for a fixed prime $p$, and show by example that this additivity can fail more generally for nilpotent groups. We also determine $δ(G)$ for several important classes of non-abelian $p$-groups, {\it viz.} VZ groups, Camina groups, and metacyclic groups; and compute $δ(G)$ for all groups of order $p^6$ ($p\geq 5$), organized by their isoclinism family.
