Previous |  Up |  Next


LCC loop; multiplication group; inner mapping group; homomorphism
A loop $Q$ is said to be left conjugacy closed if the set $A=\{L_x/x\in Q\}$ is closed under conjugation. Let $Q$ be an LCC loop, let $\Cal L$ and $\Cal R$ be the left and right multiplication groups of $Q$ respectively, and let $I(Q)$ be its inner mapping group, $M(Q)$ its multiplication group. By Drápal's theorem [3, Theorem 2.8] there exists a homomorphism $\Lambda : \Cal L \to I(Q)$ determined by $L_x\to R^{-1}_x L_x$. In this short note we examine different possible extensions of this $\Lambda$ and the uniqueness of these extensions.
[1] Basarab A.S.: A class of LK-loops (in Russian). Mat. Issled. 120 (1991), 3-7. MR 1121425
[2] Drápal A.: Conjugacy closed loops and their multiplication groups. J. Algebra 272 (2004), 838-850. MR 2028083 | Zbl 1047.20049
[3] Drápal A.: On multiplication groups of left conjugacy closed loops. Comment. Math. Univ. Carolinae 45 (2004), 223-236. MR 2075271 | Zbl 1101.20035
[4] Goodaire E.G., Robinson D.A.: A class of loops which are isomorphic to all loop isotopes. Canad. J. Math. 34 (1982), 662-672. MR 0663308 | Zbl 0467.20052
[5] Kiechle H., Nagy G.P.: On the extension of involutorial Bol loops. Abh. Math. Sem. Univ. Hamburg 72 (2002), 235-250. MR 1941556 | Zbl 1016.20051
[6] Nagy P., Strambach K.: Loops as invariant sections in groups and their geometry. Canad. J. Math. 46 (1994), 1027-1056. MR 1295130 | Zbl 0814.20055
[7] Soikis L.R.: The special loops (in Russian). in: Voprosy teorii kvazigrupp i lup (V.D. Belousov, ed.), Akademia Nauk Moldav. SSR, Kishinyev, 1970, pp.122-131. MR 0281828
Partner of
EuDML logo