A class of commutative loops with metacyclic inner mapping groups
نویسنده
چکیده
We investigate loops defined upon the product Zm × Zk by the formula (a, i)(b, j) = ((a + b)/(1 + tf (0)f (0)), i + j), where f(x) = (sx + 1)/(tx + 1), for appropriate parameters s, t ∈ Z∗m. Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If s = 1, then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.
منابع مشابه
Small 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 کاملBruck Loops with Abelian Inner Mapping Groups
Bruck loops with abelian inner mapping groups are centrally nilpotent of class at most 2.
متن کاملSome finite groups with divisibility graph containing no triangles
Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...
متن کاملAbelian Extensions and Solvable Loops
Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated...
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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010