PERMUTATION REPRESENTATION OF A TRIANGLE GROUP

2019 ◽  
Vol 44 (2) ◽  
pp. 159-180
Author(s):  
Muhammad Sarwar Saeed ◽  
Muhammad Ashiq ◽  
Tariq Alraqad ◽  
Tahir Imran
Keyword(s):  
Permutation Representation ◽  
Triangle Group

scholarly journals Deforming the R-Fuchsian (4, 4, 4)-triangle group into a lattice

Topology ◽  
2006 ◽  
Vol 45 (6) ◽  
pp. 989-1020 ◽  
Author(s):  
Martin Deraux
Keyword(s):  
Triangle Group

scholarly journals The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

2014 ◽  
Vol 66 (2) ◽  
pp. 354-372 ◽  
Author(s):  
Ruth Kellerhals ◽  
Alexander Kolpakov
Keyword(s):  
Growth Rate ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Triangle Group ◽  
Minimal Growth ◽  
Salem Number

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

scholarly journals The theta functions of sublattices of the Leech lattice

1986 ◽  
Vol 101 ◽  
pp. 151-179 ◽  
Author(s):  
Takeshi Kondo ◽  
Takashi Tasaka
Keyword(s):  
Automorphism Group ◽  
Euclidean Space ◽  
Theta Functions ◽  
Orthogonal Basis ◽  
Permutation Representation ◽  
Dimensional Euclidean Space ◽  
Unimodular Lattice ◽  
Mathieu Group ◽  
Leech Lattice ◽  
Even Unimodular Lattice

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

scholarly journals On quasi-permutation representations of finite groups

1994 ◽  
Vol 36 (3) ◽  
pp. 301-308 ◽  
Author(s):  
J. M. Burns ◽  
B. Goldsmith ◽  
B. Hartley ◽  
R. Sandling
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Minimal Degree ◽  
Permutation Representation ◽  
Permutation Groups ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Group Situation ◽  
A Finite Group

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

S-rings over loops, right mapping groups and transversals in permutation groups

1981 ◽  
Vol 89 (3) ◽  
pp. 433-443 ◽  
Author(s):  
K. W. Johnson
Keyword(s):  
Permutation Group ◽  
Group Ring ◽  
Permutation Representation ◽  
Permutation Groups ◽  
P 75 ◽  
Regular Subgroup ◽  
Finite Set

The centralizer ring of a permutation representation of a group appears in several contexts. In (19) and (20) Schur considered the situation where a permutation group G acting on a finite set Ω has a regular subgroup H. In this case Ω may be given the structure of H and the centralizer ring is isomorphic to a subring of the group ring of H. Schur used this in his investigations of B-groups. A group H is a B-group if whenever a permutation group G contains H as a regular subgroup then G is either imprimitive or doubly transitive. Surveys of the results known on B-groups are given in (28), ch. IV and (21), ch. 13. In (28), p. 75, remark F, it is noted that the existence of a regular subgroup is not necessary for many of the arguments. This paper may be regarded as an extension of this remark, but the approach here differs slightly from that suggested by Wielandt in that it appears to be more natural to work with transversals rather than cosets.

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