Year: 1961
Type: article
Abstract: This theorein is related to a result of Hall [2, Theoremn 5.6], which states that if a group G satisfies condition above and in addition either (i') 9D1 is finite, or (i) for some i, j Cthere is at most one element of G miiapping i into j which displaces all of the letters, then G is isomorphic to the group of affine transformations, x->ax+b, a0, on a near-field. A niear-field is an algebraic system (K, +, *) consisting of a set K and two binary operations + and satisfying: (a) K(+) is an Abelian group with identity 0, (b) the nonzero elements of K form a group with respect to with identity 1, (c) x(y+z) =xy+xz for x, y, zCK, (d) Oa=O for each aCK, (e) if a, b, cCK, a #b, the equation au=bu+c, has a unique solution u in K. In [I], Gorenstein has called an independent ABA group, any group H which contains two subgroups A and B such that for xEH, either xCA, or x can be represented uniquely in the form a1ba2, a1, a2CA, 1 0 bE?B. The proof of the Theorem will consist of first showing that a doubly transitive group G, in which only the identity fixes two letters, is a special kind of independent ABA group. This is a corollary of Lemma 2. Using the structure of G as an independent ABA group, it will then be shown that, when A is Abelian, G satisfies condition (i) of Hall's theorem, from which our theorem follows at once. It should be pointed out that in the finite case Hall's result [2, Theorem 5.6] follows almost immediately from the corollary of
Author: J. L. Zemmer
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2004767779