On the Automorphism Group of a Nilpotent p‐Group
On the Automorphism Group of a Nilpotent p‐Group
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DOI：
10.1112/jlms/s2-31.2.272
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发表时间：
1985-04
影响因子：
1.2
通讯作者：
F. Menegazzo;S. Stonehewer
中科院分区：
文献类型：
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作者：
F. Menegazzo;S. Stonehewer
根据Gaschiitz [3]的一个著名定理，每个有限的？群G有一个p阶的外自同构，只要G既不是平凡的，也不是p阶的循环的。在某种意义上，这个定理已经被Zalesskii [8]推广，他证明了每个无限幂零群G都有一个p阶的外自同构。群G具有外自同构;几年后，Buckley和Wiegold [1，2]改进了ZalesskiFs的结果，确定|G外|.他们证明了|G外|= 2 'G l，如果G不是约化的（也就是说，如果G有可分子群#1），而|OutG| = 2个|B|其中B是G的一个基本子群，如果G是约化的.然而，很容易看出，在大多数情况下，在[8，1，2]中展示的自同构实际上是无限阶的。本文证明了，除了明显的例外（单位群，p阶循环群和秩小于1的可分交换群），每个幂零π-群G都有外π-自同构;而且，如果G的基本子群π是无限的，则|G外|有一个基数为2 'BI的初等阿贝尔子群。
According to a celebrated theorem of Gaschiitz [3], every finite/?-group Ghas an outer automorphism of order p, provided only that G is neither trivial nor cyclic of order p. In a sense, this theorem has been generalized by Zalesskii [8], who showed that every infinite nilpotent/?-group G has an outer automorphism; and some years later Buckley and Wiegold [1, 2] improved upon ZalesskiFs result, determining| Out G|. They showed that| Out G|= 2'G l if G is not reduced (that is if G has divisible subgroups# 1), while| OutG|= 2| B|, where B is a basic subgroup of G, if G is reduced. However, it is easily seen that in most cases the automorphisms exhibited in [8, 1, 2] are in fact of infinite order. In this paper we prove that, with the obvious exceptions (the identity group, cyclic groups of order p and divisible abelian groups of rank smaller than/>—1), every nilpotent/7-group G has an outer/^-automorphism; moreover, if a basic subgroup Bof G is infinite, then| Out G| has an elementary abelian/^-subgroup of cardinality 2'BI.