Character Theory of Finite Groups

Abstract We present a detailed account of the properties of $$\text {twister}$$ twister s and their generalizations, $$\text {FC set}$$ FC set s, which are essential ingredients of the orbifold deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one, and if so, to identify the twist group and the original model. The close analogy with the character theory of finite groups is discussed, and its origin explained.

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Word problems for finite nilpotent groups

Archiv der Mathematik ◽

10.1007/s00013-020-01504-w ◽

2020 ◽

Vol 115 (6) ◽

pp. 599-609

Author(s):

Rachel D. Camina ◽

Ainhoa Iñiguez ◽

Anitha Thillaisundaram

Keyword(s):

Finite Group ◽

Nilpotent Group ◽

Finite Groups ◽

Irreducible Character ◽

Nilpotent Groups ◽

Nilpotency Class ◽

Character Degrees ◽

Character Theory ◽

Odd Order ◽

Related Group

AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

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