FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER.
FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER.
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DOI：
10.1073/pnas.45.4.578
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发表时间：
1959-04
影响因子：
11.1
通讯作者：
John G. Thompson
中科院分区：
文献类型：
--
作者：
John G. Thompson
1. Introduction.-A theorem of Frobenius1 states that if G is a finite group, H is a subgroup of G which is its own normalizer and has trivial intersection with each of its conjugates, then the set of elements of G which do not lie in any conjugate of H, together with the identityelement, forms a normal subgroup N of G, and an element h e H, h 5 $1, induces an automorphism of N which leaves only the identity element fixed. Conversely, if a group N possesses a fixed-point-free automorphism 0f of prime order, then the holomorph (split extension) of N by Ia} is a group G withIo} in the role of H. Hence, groups Nwhich can arise in Frobenius' theorem are precisely those groups with fixed-point-free automorphisms of prime order. 2 Frobenius' theorem left unanswered the more detailed analysis of the possible groups H and N whichcan arise in this manner. Burnside3 showed that the Sylow subgroups of H must be cyclic or generalized quaternion, and incorrectly stated that H must be nilpotent, an error first pointed out by Zassenhaus and discussed later by Sah. 4 More recently, Suzuki5 has given a complete classification of all finite groups with cyclic Sylow subgroups for all odd primes and with 2-Sylow subgroups which are cyclic, generalized quaternion, or dihedral, so the structure of H is known.
