Space Groups and Groups of Prime Power Order VI. a Bound to the Dimension of a 2-Adic Space Group with Fixed Coclass
Space Groups and Groups of Prime Power Order VI. a Bound to the Dimension of a 2-Adic Space Group with Fixed Coclass
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空间群和主力阶群 VI。
DOI：
10.1112/jlms/s2-34.3.417
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发表时间：
1986
影响因子：
1.2
通讯作者：
W. Plesken
中科院分区：
文献类型：
--
作者：
C. Leedham;S. Mckay;W. Plesken
在文[2]中，我们证明了文[1]中关于奇素数的猜想E。更精确地说，我们证明了维数为pp~\p-\）的唯一p-adic空间群的余类至少是p-\。如在[2]中一样，n维p进空间群R是秩为n的p进整数Zp上的自由模T被有限p进空间群R扩张。群P忠实地作用于T。此外，如果点群P不是平凡的，并且T的ZpP-子格是按包含线性序的，则称R是唯一的。R的上类是R的有限因子群X的上类的极限，定义为vp（\X\）-类（X），其中vp是Z的p-adic指数赋值。在本文中，我们证明了相应的界限的情况下，p= 2。
In a previous paper [2], we proved Conjecture E of [1] for odd primes. More precisely we proved that the coclass of a uniserial p-adic space group of dimension pp~\p-\) is at least p-\. As in [2] an «-dimensional/>-adic space group R is an extension of a free module T over the p-adic integers Zp of rank n by a finite/?-group P which acts faithfully on T. Moreover R is called uniserial if the point group P is not trivial and the ZpP-sublattices of Tare linearly ordered by inclusion. The coclass of R is the limit of the coclasses of the finite factor groups X of R, which are defined as vp (\X\)—class (X), where vp is thep-adic exponential valuation of Z. In this paper we prove the corresponding bound in the case where p= 2.