Inner mappings of Bruck loops
نویسنده
چکیده
K-loops have their origin in the theory of sharply 2-transitive groups. In this paper a proof is given that K-loops and Bruck loops are the same. For the proof it is necessary to show that in a (left) Bruck loop the left inner mappings L(b)L(a)L(ab)−" are automorphisms. This paper generalizes results of Glauberman[3], Kist[8] and Kreuzer[9].
منابع مشابه
Bruck Loops with Abelian Inner Mapping Groups
Bruck loops with abelian inner mapping groups are centrally nilpotent of class at most 2.
متن کاملMoufang Loops with Commuting Inner Mappings
We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most two. 6-divisible Moufang loops with commuting inner mappings have nilpotency class at most two. There is a Moufang loop of order 2 with commuting inner mappings and of nilpotency class three.
متن کاملThe Structure of Automorphic Loops
Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman’s work on uniquely 2-divisible Moufang loops) and the associated L...
متن کاملSmall Loops of Nilpotency Class Three with Commutative Inner Mapping Groups
Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Csörgő type. In order to obtain small loops of Csörgő type, we expand our programme from Explicit constructions of loops with commuting inner mappings, European J. Combin. 29 (2008), 1662–1681, and analyze the foll...
متن کامل1 8 A ug 2 00 9 On Bruck Loops of 2 - power Exponent ∗
We classify " nice " loop envelopes to Bruck loops of 2-power exponent under the assumption that every nonabelian simple section of G is either passive or isomorphic to PSL2(q), q − 1 ≥ 4 a 2-power. The hypothesis is verified in a separate paper. This paper is a continuation of the work by Aschbacher, Kinyon and Phillips on finite Bruck loops [AKP]. In [BS3] we will apply these results and get ...
متن کامل