有限群中不同元素阶的数量

中文摘要 AI 辅助

对于有限群$G$，令$\omega(G)$表示其谱，即其元素阶的集合，并设$\eta(G)=|\omega(G)|$，即不同元素阶的数量。集合$\omega(G)$已被深入研究，但其基数作为不变量本身却很少受到系统关注。我们发展了$\eta$的理论。在描述了它在子群、商群、截断、直积和Frobenius扩张下的行为之后，我们证明了双边界$1+\sum_{p\mid|G|}v_p(\exp G)\le\eta(G)\le\tau(\exp G)$，其中下界等式刻画了CP-群，即所有元素都具有素数幂阶的群，而上界等式刻画了其谱实现指数每个因子的群；幂零群位于上端，给出$\eta(G)=\tau(\exp G)$。我们还证明了$\eta(G)\le\tau(|G|)$，等式仅对循环群成立，以及$\eta(G)\ge\pi(|G|)+1$，等式仅当每个非平凡元素具有素数阶时成立。我们证明$\eta(G)\le3$强制可解性，其论证仅依赖于Burnside定理，并结合通过谱识别$A_5$，得出$A_5$是唯一具有$\eta(G)=4$的不可解群。对于对称群和交错群，我们记录了通过单个算术函数给出谱成员资格的封闭判据的自包含证明，将相关的计数序列置于当前框架中，并确定了固定阶的所有群上$\eta$的极值。

英文摘要

For a finite group $G$ let $ω(G)$ denote its spectrum, the set of orders of its elements, and set $η(G)=|ω(G)|$, the number of distinct element orders. The set $ω(G)$ has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of $η$. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound $1+\sum_{p\mid|G|}v_p(\exp G)\leη(G)\leτ(\exp G)$, in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving $η(G)=τ(\exp G)$. We also prove $η(G)\leτ(|G|)$ with equality only for cyclic groups, and $η(G)\geπ(|G|)+1$ with equality only when every nontrivial element has prime order. We show that $η(G)\le3$ forces solvability by an argument resting only on Burnside's theorem, and, combining this with the recognition of $A_5$ by its spectrum, that $A_5$ is the unique non-solvable group with $η(G)=4$. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of $η$ over all groups of a fixed order.

发表机构

• St. Xavier’s College, Ranchi affiliated to Ranchi University（兰契圣泽维尔学院（隶属兰契大学））
• Gossner College, Ranchi（兰契戈斯纳学院）

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