| 余大鹏,张志让,吕恒.关于广义Dedekind群[J].数学年刊A辑,2011,32(3):331~338 |
| 关于广义Dedekind群 |
| On the Generalized Dedekind Groups |
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| DOI： |
| 中文关键词: Dedekind群, 幂零群, 局部幂零群 |
| 英文关键词:Dedekind groups, Nilpotent groups, Locally nilpotent groups |
| 基金项目:国家自然科学基金 (No.11071229), 数学天元基金 (No.10926030), 青年基金 (No.11001226) 和西南大学专项基金 (No.XDJK2009C068) |
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| 中文摘要: |
| 如果群$G$的任意循环子群$H$满足$|H^{G}:H| \leq p$, 其中$p$是素数, 那么称$G$是$C^*(p)$-群.
若群$G$是有限$C^*(p)-p$-群, 则当$p>3$时, 该群的幂零类至多为2; 若$p=3$, 该群的幂零类至多为3, 而且当cl$(G)=3$时,
$\exp(G)=9$; 同时,若$G$与任意有限$C^*(p)-p$-群$G \times K$直积是$C^*(p)-p$-群$G \times K$, 则$G$是初等阿贝尔$p$-群.
最后还对局部幂零的$C^*(p)$-群进行了探讨. |
| 英文摘要: |
| A group $G$ is a $C^*(p)$-group, if each cyclic subgroup $H$ of $G$ satisfies $|H^{G}:H| \leq p$.
The authors prove that if $G$ is a finite $C^*(p)$-group, then the nilpotent class of $G$ is at most 2 when $p>3$;
if $p=3$ and the nilpotent class of $G$ is 3, then $p=3$ and $\exp(G)=9$.
Furthermore, if for any finite $C^*(p)-p$-group $K$, the direct product $G \times K$ is also a $C^*(p)-p$-group,
then $G$ is an elementary Abelian $p$-group. Finally, the locally nilpotent $C^*(p)$-group is discussed. |
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