Spin representations of twisted central products of double covering finite groups and the case of permutation groups

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.

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Some finite groups with few defining relations

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700023090 ◽

1985 ◽

Vol 38 (2) ◽

pp. 230-240 ◽

Cited By ~ 5

Author(s):

B. H. Neumann

Keyword(s):

Finite Groups ◽

Direct Products ◽

Cyclic Groups ◽

Defining Relations ◽

Central Products

AbstractSome new classes of finite groups with zero deficiency presentations, that is to say presentations with as few defining relations as generators, are exhibited. The presentations require 3 generators and 3 defining relations; the groups so presented can also be generated by 2 of their elements, but it is not known whether they can be defined by 2 relations in these generators, and it is conjectured that in general they can not. The groups themselves are direct products or central products of binary polyhedral groups with cyclic groups, the order of the cyclic factor being arbitrary.

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Quantum reflections, random walks and cut-off

International Journal of Mathematics ◽

10.1142/s0129167x1850101x ◽

2018 ◽

Vol 29 (14) ◽

pp. 1850101 ◽

Cited By ~ 1

Author(s):

Amaury Freslon

Keyword(s):

Random Walks ◽

Quantum Groups ◽

Finite Groups ◽

Wreath Products ◽

Conjugacy Classes ◽

Permutation Groups ◽

Reflection Groups ◽

Quantum Reflection ◽

Quantum Analogue

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of reflections. We also study random walks on quantum reflection groups and more generally on free wreath products of finite groups by quantum permutation groups.

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More on finite groups with few defining relations

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700030421 ◽

1989 ◽

Vol 46 (1) ◽

pp. 132-136 ◽

Cited By ~ 1

Author(s):

J. L. Mennicke ◽

B. H. Neumann

Keyword(s):

Finite Groups ◽

Cyclic Groups ◽

Defining Relations ◽

Central Products

AbstractCertain central products of the binary polyhedral groups with finite cyclic groups are here shown to have presentations with two generators and two defining relations; this disproves a conjecture of the second author, stated in J. Austral. Math. Soc. Ser. A 38 (1985), 230–240.

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The primitive permutation groups of degree less than 1000

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064793 ◽

1988 ◽

Vol 103 (2) ◽

pp. 213-238 ◽

Cited By ~ 30

Author(s):

John D. Dixon ◽

Brian Mortimer

Keyword(s):

Finite Groups ◽

Primitive Group ◽

Simple Groups ◽

Permutation Groups ◽

Finite Simple Groups ◽

Primitive Groups ◽

Concrete Form

Our object is to describe all of the-primitive permutation groups of degree less than 1000 together with some of their significant properties. We think that such a list is of interest in illustrating in concrete form the kinds of primitive groups which arise, in suggesting conjectures about primitive groups, and in settling small exceptional cases which often occur in proofs of theorems about permutation groups. The range that we consider is large enough to allow examples of most of the types of primitive group to appear. Earlier lists (of varying completeness and accuracy) of primitive groups of degree d have been published by: C. Jordan (1872) [21] ford≤ 17, by W. Burnside (1897) [5] ford≤ 8, by Manning (1929) [34–38] ford≤ 15, by C. C. Sims (1970) [45] ford≤ 20, and by B. A. Pogorelev (1980) [42] ford≤ 50. Unpublished lists have also been prepared by C. C. Sims ford≤ 50 and by Mizutani[41] ford≤ 48. Using the classification of finite simple groups which was completed in 1981 we have been able to extend the list much further. Our task has been greatly simplified by the detailed information about many finite simple groups which is available in theAtlas of Finite Groupswhich we will refer to as theAtlas[8].

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The Double Transitivity of a Class of Permutation Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1965-047-9 ◽

1965 ◽

Vol 17 ◽

pp. 480-493 ◽

Cited By ~ 5

Author(s):

Ronald D. Bercov

Keyword(s):

Finite Groups ◽

Abelian Groups ◽

Transitive Group ◽

Primitive Group ◽

Permutation Groups ◽

Regular Subgroup

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).

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