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@PHDTHESIS{Kirschmer:50695,
author = {Kirschmer, Markus},
othercontributors = {Nebe, Gabriele},
title = {{F}inite symplectic matrix groups},
address = {Aachen},
publisher = {Publikationsserver der RWTH Aachen University},
reportid = {RWTH-CONV-113228},
pages = {151 S.},
year = {2009},
note = {Aachen, Techn. Hochsch., Diss., 2009},
abstract = {The finite subgroups of GL(m, Q) are those subgroups that
fix a full lattice in $Q^m$ together with some positive
definite symmetric form. A subgroup of GL(m, Q) is called
symplectic, if it fixes a nondegenerate skewsymmetric form.
Such groups only exist if m is even. A symplectic subgroup
of GL(2n, Q) is called maximal finite symplectic if it is
not properly contained in some finite symplectic subgroup of
GL(2n, Q). This thesis classifies all conjugacy classes of
maximal finite symplectic subgroups of GL(2n, Q) up to
2n=22. The natural representation of a maximal finite
symplectic matrix group is a sum of pairwise nonisomorphic
rationally irreducible representations that yield maximal
finite symplectic matrix groups. Thus, it suffices to
classify the (conjugacy classes of) symplectic irreducible
maximal finite (s.i.m.f.) matrix groups. One can proceed as
in the classification of the maximal finite subgroups of
GL(m, Q). Each s.i.m.f. matrix group is the full
automorphism groups of some lattice with respect to a
symmetric positive definite form and a skewsymmetric form. A
symplectic matrix group is called symplectic imprimitive if
it is contained (up to conjugacy) in a wreath product of
some symplectic matrix group. The symplectic imprimitive
matrix groups can be constructed by the classification of
the s.i.m.f. subgroups of smaller dimension. Further they
can easily be recognized by orthogonal decompositions of
invariant lattices. Thus we only have to classify the
symplectic primitive irreducible maximal finite (s.p.i.m.f.)
matrix groups. The concept of primitivity has some important
consequences. The restriction of the natural representation
of a s.p.i.m.f. matrix group to a normal subgroup is a
multiple of a single rationally irreducible representation.
This implies that there are only finitely many possibilites
for the generalized Fitting subgroup of a s.p.i.m.f. matrix
group G < GL(2n, Q). Moreover, the list of candidates only
depends on n. The possible Fitting subgroups are given by a
theorem of Hall. The possible layers (central products of
quasisimple groups) can be taken from the ATLAS of finite
simple groups. A useful tool for the classification of all
s.p.i.m.f. matrix groups G is the so-called generalized
Bravais group B(N). If N is normal in G, so is B(N).
Furthermore, it is possible to classify all s.i.m.f.
supergroups of a given irreducible matrix group U, provided
that the commuting algebra of U is a field. Moreover, some
infinite families of s.i.m.f. matrix groups are constructed.
In particular, all s.i.m.f. subgroups of GL(p-1, Q) and
GL(p+1, Q) whose orders are divisible by a prime p>=5 are
determined.},
keywords = {Matrizengruppe (SWD) / Ganzzahliges Gitter (SWD) /
Ganzzahlige Darstellungstheorie (SWD) / Algebraische
Zahlentheorie (SWD)},
cin = {110000 / 114710},
ddc = {510},
cid = {$I:(DE-82)110000_20140620$ / $I:(DE-82)114710_20140620$},
shelfmark = {20H20 * 20C10 * 20G30 * 11E12 * 20-04},
typ = {PUB:(DE-HGF)11},
urn = {urn:nbn:de:hbz:82-opus-27584},
url = {https://publications.rwth-aachen.de/record/50695},
}
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