# zbMATH — the first resource for mathematics

On collineation groups generated by Bol reflections. (English) Zbl 0793.51001
A loop $$(L,o)$$ is a (left) Bol loop if the identity $x o(y o(x o z))=(x o(y o x))o z$ holds for all $$x$$, $$y$$, $$z\in L$$. In a 3-net, $$N$$, one can arbitrarily specify the three pencils as being horizontal $$(H)$$, vertical $$(V)$$ and transversal $$(T)$$. For a point $$P\in N$$ let $$P_ h$$, $$P_ v$$ and $$P_ t$$ denote the horizontal, vertical and transversal lines through $$P$$, respectively. A 3-net which can be coordinatized by a Bol loop is called a Bol 3-net and satisfies the condition: For every vertical line $$g$$, the mapping $$\sigma_ g: N\to N$$ defined by $\sigma_ g(P)=(P_ h\cap g)_ t\cap(P_ t\cap g)_ h$ is a collineation of $$N$$, which preserves the pencil $$V$$ and interchanges the pencils $$H$$ and $$T$$. These collineations of the 3-net are called Bol reflections.
This paper investigates the connection between geometric transformation groups and algebraic invariants for Bol 3-nets. The geometric tools are Bol reflections and the group of collineations that they generate. Modulo a subgroup in the left nucleus of the coordinate loop, the corresponding algebraic invariant becomes the group generated by all left multiplications. This group admits a second interpretation as the subgroup of projectivities generated by perspectivities whose carriers are vertical lines. The authors prove that every such projectivity comes from a collineation generated by Bol reflections. They also examine a wide class of Bol 3-nets where this representation is faithful. Finally, the Bol condition is reformulated as a configuration condition for projective planes, which leads to characterizations of Bol planes and Moufang planes.

##### MSC:
 51A20 Configuration theorems in linear incidence geometry 51A25 Algebraization in linear incidence geometry 51F15 Reflection groups, reflection geometries
Full Text:
##### References:
  M. A. AKIVIS and A. M. SHELEKHOV, ?Foundations of the Theory of Webs,?University of Kalinin, Kalinin 1981 [in Russian]. · Zbl 0475.53016  A. BARLOTTI and K. STRAMBACH, The Geometry of Binary Systems,Advances Math. 49 (1983), 1-105. · Zbl 0518.20064 · doi:10.1016/0001-8708(83)90013-0  V. D. BELOUSOV, ?Algebraic Nets and Quasigroups,? Acad. Sciences of M. S. S. R., Kishinev, 1971 [in Russian].  R. P. BURN, Bol Quasi-Fields and Pappus’ Theorem,Math. Z. 105 (1968), 351-364. · Zbl 0159.49803 · doi:10.1007/BF01110297  M. FUNK, Octagonality Conditions in Projective and Affine Planes,Geometriae Dedicata 28 (1988), 53-75. · Zbl 0673.51001 · doi:10.1007/BF00147800  G. GLAUBERMANN, On Loops of Odd Order,J. Algebra 1 (1964), 374-396. · Zbl 0123.01502 · doi:10.1016/0021-8693(64)90017-1  K.-H. HOFMANN and K. STRAMBACH, Topological and analytical loops, in: ?Quasigroups and Loops: Theory and Applications? (ed.: O. CHEIN et al.), Heldermann, Berlin 1990, pp. 205-262. · Zbl 0747.22004  D. R. HUGHES and F. C. PIPER, ?Projective Planes,? Springer, Berlin Heidelberg New York 1973.  M. J. KALLAHER, Projective Planes over Bol Quasi-fields,Math. Z. 109 (1969), 53-65. · Zbl 0174.23602 · doi:10.1007/BF01135573  H. LÜNEBURG, ?Translation Planes,? Springer, Berlin Heidelberg New York 1980.  P. O. MIHEEV and L. V. SABININ, ?The Theory of Smooth Bol Loops,? Friendship of Nations University, Moskow 1985.  P. T. NAGY and K. STRAMBACH, Loops and symmetric spaces, to appear. · Zbl 0909.20052  P. T. NAGY and K. STRAMBACH, Holonomy groups in loop systems, to appear.  G. PICKERT, ?Projektive Ebenen,? Springer, Berlin Heidelberg New York, second edition 1975.  D. A. ROBINSON, Bol loops,Trans. Amer. Math. Soc. 123 (1966), 341-353. · doi:10.1090/S0002-9947-1966-0194545-4  D. A. ROBINSON, A loop-theoretical study of right-sided quasigroups,Ann. Soc, Sei. Bruxelles 93 (1979), 7-16. · Zbl 0414.20058  D. A. ROBINSON, The Bryant-Schneider group,Ann. Soc. Sci. Bruxelles 94 (1980), 69-81. · Zbl 0448.20060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.