On the Automorphism Group of A 2-Group
On the Automorphism Group of A 2-Group
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DOI：
10.1112/plms/s3-26.2.207
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发表时间：
1973-03
影响因子：
1.8
通讯作者：
T. Hawkes
中科院分区：
文献类型：
--
作者：
T. Hawkes
主要定理。设P是一个有限的2-群，H是P的一个自同构群，固定P的对合，则H/O2（H）与某些二面体群的直积的子群同构;这些二面体群的阶具有2 pn的形式，其中对奇素数幂pn的唯一限制是每个幂都应整除2s+ 1形式的某个整数。特别地，H是可解的，证明的方法是在假设H是可解的情况下，先研究H的结构。在§ 1中，我们开始讨论一个有用的约化定理和它所产生的所谓超特殊群。然后，我们研究了H的限制，当H在§ 2中是循环的，在§ 3中是幂零的，在§ 4中是奇数阶的，在§ 5中是可解的。最后，在§ 6中我们证明了H确实总是可解的。
MAIN THEOREM. Let P be a finite 2-group and let H be a group of automorphisms of P fixing the involutions of P. Then H/O2 (H) is isomorphic with a subgroup of a direct product of certain dihedral groups; the orders of these dihedral groups have the form 2pn, where the only restriction on the odd prime powers pn is that each should divide some integer of the form 2s+ 1. In particular, H is soluble.The method of proof is to study the structure of H first under the assumption that it is soluble. We begin in § 1 with a useful reduction theorem and a discussion of the so-called ultraspecial groups it yields. We then investigate restrictions on H when H is cyclic in § 2, nilpotent in § 3, of odd order in § 4, and soluble in § 5. Finally, in § 6 we prove that H is indeed always soluble.