## ON CONJUGACY CLASSES AND DERIVED LENGTH (905)

### BibTeX

@MISC{Adan-bante905onconjugacy,

author = {Edith Adan-bante},

title = {ON CONJUGACY CLASSES AND DERIVED LENGTH},

year = {905}

}

### OpenURL

### Abstract

Abstract. Let G be a finite group and A, B and D be conjugacy classes of G with D ⊆ AB = {xy | x ∈ A, y ∈ B}. Denote by η(AB) the number of distinct conjugacy classes such that AB is the union of those. Set CG(A) = {g ∈ G | x g = x for all x ∈ A}. If AB = D then CG(D)/(CG(A) ∩ CG(B)) is an abelian group. If, in addition, G is supersolvable, then the derived length of CG(D)/(CG(A) ∩ CG(B)) is bounded above by 2η(AB). 1.

### Citations

7 | Products of characters and finite p-groups
- Adan-Bante
(Show Context)
Citation Context ...[7] has an affirmative answer. We would like also to point out that there are several examples of “dual results” between products of conjugacy classes and products of character. For example, see [9], =-=[2]-=- and [8], [3] and [5]. However not every result in products of characters has a “dual” result in conjugacy classes, see for instance Section 3. Let Z1(G) = Z(G) be the center of the group G and by ind... |

6 | Products of characters and derived length
- Adan-Bante
(Show Context)
Citation Context ... a p-group, χ is an irreducible character of G and χ(1) = p n , then the product χχ of χ and χ has at least 2n(p − 1) + 1 distinct irreducible constituents, i.e η(χχ) ≥ 2n(p − 1) + 1. In Theorem A of =-=[1]-=- is proved that if χ is an irreducible character of a solvable group G with χ(1) > 1, then χ(1) has at most η(χχ) −1 different prime factors. If, in addition, G is supersolvable, then χ(1) has at most... |

6 |
On the normal number of prime factors of p−1 and some related problems concerning Euler’s φ-function, Quart
- Erdős
- 1935
(Show Context)
Citation Context ...f P. Then P contains just one nontrivial conjugacy class A of G, so η(AA −1 ) = 2. Also P = CG(A), and thus P contains |G/CG(A)| = p − 1. This is obviously unboundedly large, and by a result of Erdos =-=[10]-=-, it has unboundedly many prime factors. Remark. Let G be the group as in the previous proof. Let λ be an irreducible character of N. Then λ is a linear character and the induced character λ G is an i... |

5 | Conjugacy classes and finite p-groups
- Adan-Bante
(Show Context)
Citation Context ...e answer. We would like also to point out that there are several examples of “dual results” between products of conjugacy classes and products of character. For example, see [9], [2] and [8], [3] and =-=[5]-=-. However not every result in products of characters has a “dual” result in conjugacy classes, see for instance Section 3. Let Z1(G) = Z(G) be the center of the group G and by induction define the i-c... |

4 | Homogeneous products of conjugacy classes - Adan-Bante |

2 | Derived length and products of conjugacy classes
- Adan-Bante
(Show Context)
Citation Context ...constants c and d such that for any solvable group G, any irreducible characters χ, ψ and θ such that θ is a constituent of χψ, we have that dl(Ker(θ)/(Ker(χ) ∩ Ker(ψ))) ≤ cη(χψ) + d. In Theorem A of =-=[7]-=- is proved that given any supersolvable group G and any conjugacy class A, we have that dl(G/CG(A)) ≤ 2η(AA −1 ) − 1. We conjecture in [7] that there exist universal constants q and r such that for an... |

2 |
On nilpotent groups and conjugacy classes, preprint
- Adan-Bante
(Show Context)
Citation Context ...an affirmative answer. We would like also to point out that there are several examples of “dual results” between products of conjugacy classes and products of character. For example, see [9], [2] and =-=[8]-=-, [3] and [5]. However not every result in products of characters has a “dual” result in conjugacy classes, see for instance Section 3. Let Z1(G) = Z(G) be the center of the group G and by induction d... |

2 |
An analogy between products of two conjugacy classes and products of two irreducible characters in finite groups
- Arad, Fisman
- 1987
(Show Context)
Citation Context ...e of [7] has an affirmative answer. We would like also to point out that there are several examples of “dual results” between products of conjugacy classes and products of character. For example, see =-=[9]-=-, [2] and [8], [3] and [5]. However not every result in products of characters has a “dual” result in conjugacy classes, see for instance Section 3. Let Z1(G) = Z(G) be the center of the group G and b... |

1 |
On nilpotent groups and conjugacy classes, to appear Houston
- Adan-Bante
(Show Context)
Citation Context ... has an affirmative answer. We would like to point out that there are several examples of “dual results” between products of conjugacy classes and products of character. For example, see [9], [2] and =-=[8]-=-, [3] and [5]. However not every result in products of characters has a “dual” result in conjugacy classes, see for instance Section 3. In Section 4 we provide an example of a property in a conjugacy ... |