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c-Extensions of the F4(2)-building
We construct four geometries E 1,…, E 4 with the diagram such that any two elements of type 1 are incident to at most one common element of type 2 and three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The autom...
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| Published in: | Discrete mathematics 2003-03, Vol.264 (1), p.91-110 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Citations: | Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We construct four geometries
E
1,…,
E
4
with the diagram
such that any two elements of type 1 are incident to at most one common element of type 2 and three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The automorphism group
E
i
of
E
i
is flag-transitive, isomorphic to
2E
6(2)
:
2
,
3·
2E
6(2)
:
2
,
2
26
:
F
4(2)
and
E
6(2)
:
2
, for
i=1,2,3 and 4. We calculate the suborbit diagram of the collinearity graph of
E
i
with respect to the action of
E
i
. By considering the elements in
E
i
fixed by a subgroup
T
i
of order 3 in
E
i
we obtain four geometries
T
1,…,
T
4
with the diagram
on which
C
E
i
(
T
i
) induces flag-transitive action, isomorphic to
U
6(2)
:
2
,
3·U
6(2)
:
2
,
2
14
:
Sp
6(2)
and
L
6(2)
:
2
for
i=1,2,3 and 4. Next, by considering the elements fixed by a subgroup
S
i
of order 7 in
E
i
we obtain four geometries with the diagram
on which
C
E
i
(
S
i
) induces flag-transitive action isomorphic to
L
3(4)
:
2
,
3·L
3(4)
:
2
,
2
8
:
L
3(2)
and
(L
3(2)
×
L
3(2))
:
2
, for
i=1,2,3 and 4. |
|---|---|
| ISSN: | 0012-365X 1872-681X |
| DOI: | 10.1016/S0012-365X(02)00554-X |