scholarly journals Non-abelian anyons and some cousins of the Arad-Herzog conjecture

2021 ◽  
Author(s):  
Matthew Buican ◽  
Linfeng Li ◽  
Rajath Radhakrishnan
Keyword(s):  
Conjugacy Class ◽  
Gauge Theories ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Gauge Groups ◽  
Physical Intuition

Abstract Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for non-abelian anyons in 2 + 1-dimensional discrete gauge theories. Thinking in this way also suggests closely related statements about finite simple groups and their associated discrete gauge theories. We prove these statements and provide some physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with non-abelian finite simple gauge groups provides a non-trivial check of the AH conjecture.

scholarly journals Finite simple groups with short Galois orbits on conjugacy classes

2020 ◽  
Vol 544 ◽  
pp. 151-169
Author(s):  
Victor Bovdi ◽  
Thomas Breuer ◽  
Attila Maróti
Keyword(s):  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Galois Orbits

Finite groups with at most six vanishing conjugacy classes

2021 ◽  
pp. 2250076
Author(s):  
Sajjad M. Robati ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Formula ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

A note on powers in simple groups

1997 ◽  
Vol 122 (1) ◽  
pp. 91-94 ◽  
Author(s):  
JAN SAXL ◽  
JOHN S. WILSON
Keyword(s):  
Conjugacy Class ◽  
Simple Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Groups ◽  
Similar Combination

In [7], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S. The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [5] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S=R3∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.

The Inductive Blockwise Alperin Weight Condition for PSp4(q)

2019 ◽  
Vol 26 (03) ◽  
pp. 361-386 ◽  
Author(s):  
Conghui Li ◽  
Zhenye Li
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Weight Condition ◽  
Brauer Characters ◽  
A Finite Group ◽  
Alperin Weight Conjecture

Let G be a finite group and ℓ be any prime dividing [Formula: see text]. The blockwise Alperin weight conjecture states that the number of the irreducible Brauer characters in an ℓ-block B of G equals the number of the G-conjugacy classes of ℓ-weights belonging to B. Recently, this conjecture has been reduced to the simple groups, which means that to prove the blockwise Alperin weight conjecture, it suffices to prove that all non-abelian simple groups satisfy the inductive blockwise Alperin weight condition. In this paper, we verify this inductive condition for the finite simple groups [Formula: see text] and non-defining characteristic, where q is a power of an odd prime.

scholarly journals Lie theory of finite simple groups and the Roth property

2017 ◽  
Vol 163 (2) ◽  
pp. 301-340 ◽  
Author(s):  
J. LÓPEZ PEÑA ◽  
S. MAJID ◽  
K. RIETSCH
Keyword(s):  
Finite Group ◽  
Dimensional Subspace ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Killing Form ◽  
Dihedral Groups ◽  
Invariant Vectors ◽  
A Finite Group ◽  
Conjugation Representation

AbstractIn noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

scholarly journals Conjugacy in groups with dihedral 3-normalisers

1977 ◽  
Vol 18 (2) ◽  
pp. 167-173 ◽  
Author(s):  
N. K. Dickson
Keyword(s):  
Conjugacy Class ◽  
The Other ◽  
Simple Groups ◽  
Character Theory ◽  
Finite Simple Groups ◽  
Strong Connection

Much work has been carried out on the classification of finite simple groups in terms of the structures of centralisers of involutions. However, it is sometimes the case that these classification results cannot be applied to particular problems even although information is available about one conjugacy class of involutions. The trouble is that information about the other classes can be almost non-existent. In this paper we deal with a situation where character theory can be employed to give a strong connection between the orders of centralisers of different classes of involutions, enabling information about one class to be used to give information about other classes. We prove the following result.

scholarly journals Towards the classification of finite simple groups with exactly three or four supercharacter theories

2018 ◽  
Vol 11 (05) ◽  
pp. 1850096 ◽  
Author(s):  
A. R. Ashrafi ◽  
F. Koorepazan-Moftakhar
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Compatibility Conditions ◽  
Irreducible Characters ◽  
Supercharacter Theory ◽  
A Finite Group

A supercharacter theory for a finite group [Formula: see text] is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions. The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.

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