# The collineation groups of the finite affine and projective planes with four lines through each point

The collineation groups of the finite affine and projective planes with four lines through each... The collineation groups oE the finite affine and projective planes with four lines through each point Dedicated to WILHELM BLASCHKE on his seventieth birthday By H. S. M. COXETER in Toronto The group of all collineations in the finite projective plane PG(2, 3) is the simple group LF(3, 3) of order 5616. BRAHANA (1930, p.531) gave two matrices which generate this group, but did not attempt to find a complete set of defining relations. In terms of a different pair of generators, such relations are now found to be (1) 8 = T3 = (8T)4 = (S2T)4 = (8 T)3 = E, 2 2 2 2 (T8 T)28 = 8 (T8 T)2. These are established by enumerating the 26 cosets of the subgroup T3 = U3 = (TU)4 = E, (TUT)2U = U(TUT)2 (2) (where U = 8 ), which is the Hessian group of order 216, i. e., the group of projective collineations leaving invariant the nine inflexions of the general cubic curve in the complex projective plane. This, in turn, has a 8ubgroup of index 9: (3) U3 = E, U V U = V U V (where V = T-l U T), which is the binary http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

# The collineation groups of the finite affine and projective planes with four lines through each point

, Volume 20 (4) – Aug 1, 1956
13 pages      /lp/springer-journals/the-collineation-groups-of-the-finite-affine-and-projective-planes-1ZfjMBchY0
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF03374555
Publisher site
See Article on Publisher Site

### Abstract

The collineation groups oE the finite affine and projective planes with four lines through each point Dedicated to WILHELM BLASCHKE on his seventieth birthday By H. S. M. COXETER in Toronto The group of all collineations in the finite projective plane PG(2, 3) is the simple group LF(3, 3) of order 5616. BRAHANA (1930, p.531) gave two matrices which generate this group, but did not attempt to find a complete set of defining relations. In terms of a different pair of generators, such relations are now found to be (1) 8 = T3 = (8T)4 = (S2T)4 = (8 T)3 = E, 2 2 2 2 (T8 T)28 = 8 (T8 T)2. These are established by enumerating the 26 cosets of the subgroup T3 = U3 = (TU)4 = E, (TUT)2U = U(TUT)2 (2) (where U = 8 ), which is the Hessian group of order 216, i. e., the group of projective collineations leaving invariant the nine inflexions of the general cubic curve in the complex projective plane. This, in turn, has a 8ubgroup of index 9: (3) U3 = E, U V U = V U V (where V = T-l U T), which is the binary

### Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Aug 1, 1956

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