论文标题
具有所有适当的商的组的结构实际上是nilpotent
The structure of groups with all proper quotients virtually nilpotent
论文作者
论文摘要
只是无限群体在专业组理论中起着重要作用。对于每个$ c \ geq 0 $，我们考虑更普遍的jnn $ _c $ f pribinite（或在某些地方，离散的）组，这些组不适合；这些是$ g $的组，以至于$ g $的每一个适当的商实际上都是班级-c $ nilpotent，而$ g $本身不是，而且$ g $也没有任何非平凡的Abelian普通亚组。当$ c = 1 $时，我们将获得只有非平凡的Abelian正常亚组的仅非（几乎是Abelian）组。 我们的第一个结果是，有限生成的profinite组实际上是class \ nbd $ c $ nilpotent，并且只有当时只有有限的许多子组，因为下部中央系列术语$γ_{c+1}（c+1}（k）$ open normal untorm untorm subgroups $ k $ g $ g $ $ g $。基于此，我们证明了几个结构定理。例如，我们根据上述表格$γ_{c+1}（k）$的子组来表征JNN $ _C $ f pribinite组。我们还将JNN $ _C $ F组的描述描述为几乎nilpotent profinite组的合适逆限制。为遗传性JNN $ _C $ F组建立了类似的结果，例如，我们表明，如果且仅当每个最大的有限inite Index is jnn $ _c $ _c $ f f。最后，我们提供了遗传性JNN $ _C $ F组的构造，该组用作遗传性无限群体的意见家庭。
Just infinite groups play a significant role in profinite group theory. For each $c \geq 0$, we consider more generally JNN$_c$F profinite (or, in places, discrete) groups that are Fitting-free; these are the groups $G$ such that every proper quotient of $G$ is virtually class-$c$ nilpotent whereas $G$ itself is not, and additionally $G$ does not have any non-trivial abelian normal subgroup. When $c = 1$, we obtain the just non-(virtually abelian) groups without non-trivial abelian normal subgroups. Our first result is that a finitely generated profinite group is virtually class\nbd$c$ nilpotent if and only if there are only finitely many subgroups arising as the lower central series terms $γ_{c+1}(K)$ of open normal subgroups $K$ of $G$. Based on this we prove several structure theorems. For instance, we characterize the JNN$_c$F profinite groups in terms of subgroups of the above form $γ_{c+1}(K)$. We also give a description of JNN$_c$F profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNN$_c$F groups and, for instance, we show that a Fitting-free JNN$_c$F profinite (or discrete) group is hereditarily JNN$_cF$ if and only if every maximal subgroup of finite index is JNN$_c$F. Finally, we give a construction of hereditarily JNN$_c$F groups, which uses as an input known families of hereditarily just infinite groups.
