A Note on a Bound of Adan-Bante
Let G be a finite solvable group and let χ be a nonlinear irreducible (complex) character of G. Also let \( \eta \) (χ) be the number of nonprincipal irreducible constituents of \( \upchi \overline{\upchi} \) , where \( \overline{\upchi} \) denotes the complex conjugate of χ. Adan-Bante proved...
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2014
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508141202964480 |
|---|---|
| author | Chen, Xiaoyou Чен, Хіаоіоу |
| author_facet | Chen, Xiaoyou Чен, Хіаоіоу |
| author_sort | Chen, Xiaoyou |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:25:58Z |
| description | Let G be a finite solvable group and let χ be a nonlinear irreducible (complex) character of G. Also let \( \eta \) (χ) be the number of nonprincipal irreducible constituents of \( \upchi \overline{\upchi} \) , where \( \overline{\upchi} \) denotes the complex conjugate of χ. Adan-Bante proved that there exist constants C and D such that dl (G/ ker χ) ≤ C \( \eta \) (χ) +D. In the present work, we establish a bound lower than the Adan-Bante bound for \( \eta \) (χ) > 2 |
| first_indexed | 2026-03-24T02:20:29Z |
| format | Article |
| fulltext |
UDC 512.5
Xiaoyou Chen (College Sci., Henan Univ. Technology, Zhengzhou, China)
A NOTE ON A BOUND OF ADAN-BANTE*
ОДНЕ ЗАУВАЖЕННЯ ЩОДО ГРАНИЦI АДАН-БАНТE
Let G be a finite solvable group and let χ be a nonlinear irreducible (complex) character of G. Also let η(χ) be the number
of nonprincipal irreducible constituents of χχ̄, where χ̄ denotes the complex conjugate of χ. Adan-Bante proved that there
exist constants C and D such that dl (G/ kerχ) ≤ Cη(χ) +D. In the present work, we establish a bound lower than the
Adan-Bante bound for η(χ) > 2.
Нехай G — скiнченна розв’язна група, а χ — нелiнiйний незвiдний (комплексний) характер групи G. Також нехай
η(χ) — число неголовних незвiдних складових χχ̄, де χ̄ позначає величину, комплексно спряжену до χ. Як доведено
Адан-Банте, iснують сталi C та D такi, що dl (G/ kerχ) ≤ Cη(χ) +D. В данiй роботi встановлено оцiнку нижчу,
нiж оцiнка Адан-Банте для η(χ) > 2.
Let G be a finite solvable group and χ be a nonlinear irreducible (complex) character of G. Let η(χ)
be the number of nonprincipal irreducible constituents of χχ̄, where χ̄ means the complex conjugate
of χ. In her paper [1], E. Adan-Bante utilized a key lemma to yield a bound for the derived length of
G/ kerχ. That is the following lemma.
Lemma 1. Let n > 1 be an integer and N = {1, 2, . . .} be the set of all positive integers. Define
p(n) = max{n1n2 . . . ns | n1, n2, . . . , ns ∈ N and n1 + n2 + . . .+ ns = n}.
Hence
p(n) ≤ 2n−1.
Adan-Bante’s inequality above can be improved slightly. In fact, we have the following lemma.
Lemma 1′. Let n > 1 be an integer and N = {1, 2, . . .} be the set of all positive integers. Define
p(n) = max{n1n2 . . . ns | n1, n2, . . . , ns ∈ N and n1 + n2 + . . .+ ns = n}.
Then
p(n) =
3n/3, n ≡ 0 (mod 3),
4 · 3(n−4)/3, n ≡ 1 (mod 3),
2 · 3(n−2)/3, n ≡ 2 (mod 3).
Hence
p(n) ≤ 3n/3.
Proof. By the relation of congruence, then for n ≥ 2 we have that one of the following:
n ≡ 0 (mod 3), n ≡ 1 (mod 3), or n ≡ 2 (mod 3).
* Supported by the Doctor Foundation of Henan University of Technology (2010BS048), the Project of Zhengzhou
Municipal Bureau of Science and Technology (20130790), the Key Project of Education Department of Henan Province
(14B110001), and the Project of Science and Technology Department of Henan Province (142300410133).
c© XIAOYOU CHEN, 2014
1006 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
A NOTE ON A BOUND OF ADAN-BANTE 1007
By the definition of p(n) and computation, it follows that
n = 2, p(n) = 2, n = 5, p(n) = 2 · 3,
n = 3, p(n) = 3, n = 6, p(n) = 3 · 3,
n = 4, p(n) = 4, n = 7, p(n) = 4 · 3.
We prove that the factors of p(n) are 2 or 3.
Let n = m1 +m2 + . . .+mt, t ≥ 1, such that
p(n) = m1m2 . . .mt.
We assert that
(i) mi > 1 for every i = 1, 2, . . . , t.
Otherwise, it is no loss to assume that m1 = 1. Thus,
(1 +m2)m3 . . .mt > m1m2m3 . . .mt = p(n),
a contradiction.
(ii) mi ≤ 4 for each i = 1, 2, . . . , t.
Otherwise, it is no loss to assume that m1 > 4 and then
2 · (m1 − 2) > m1.
Hence,
2 · (m1 − 2)m2m3 . . .mt > m1m2m3 . . .mt = p(n),
a contradiction.
So, mi, i = 1, 2, . . . , t, are 2 or 3 since 4 = 2 · 2 and then
p(n) = 2a3b,
where a, b are nonnegative integers and 2a+ 3b = n.
Now, since 2 · 2 · 2 < 3 · 3, it follows that the number of factor 3 in p(n) should be as many as
possible. That is,
0 ≤ a ≤ 2.
Therefore, we have that
p(n) =
3n/3, n ≡ 0 (mod 3),
4 · 3n−4/3, n ≡ 1(mod 3),
2 · 3n−2/3, n ≡ 2(mod 3).
It follows that
p(n) ≤ 3n/3.
Lemma 1′ is proved.
Utilizing the inequality p(n) ≤ 3n/3 in Adan-Bante’s proof in [1], we have that the bound of
Adan-Bante can be improved as follows.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
1008 XIAOYOU CHEN
Theorem 1. Let G be a finite solvable group and χ ∈ Irr (G), where Irr (G) denotes the set of
irreducible characters of G. Then there exists a constant c such that
dl (G/ kerχ) ≤ cη(χ) + 1.
Remark. In particular, if χ ∈ Irr(G) is faithful, we would have that dl (G) ≤ cη(χ) + 1. Note
that E. Adan-Bante has studied the finite solvable groups with η(χ) ≤ 2 in [2, 3].
Keller [4] obtained that there exist universal constants C1 and C2 such that dl (G) ≤
≤ C1 log (m(G,V)) + C2 for any finite solvable group G acting faithfully and irreducibly on a
finite vector space V. In fact, the author proved the result with log = log2, C1 = 24 and C2 = 364.
And the author says in [4] that these constants are far from being best possible. Notice that the
constants C and D in [1] are related to the constants in [4]. Actually, C = C1 log 2 + C2 + 1 and
D = 1− C1 log 2 (By the way, that Adan-Bante wrote D = 1 + C1 log 2 in [1] is a typo). Also, our
constant c =
log 3
3
C1 + C2 + 1. If η(χ) > 2, that is, η(χ) ≥ 3, and since
3 >
log 2
log 2− log 3
3
,
then we have that cη(χ) + 1 < Cη(χ) + D. So our bound is lower than Adan-Bante’s if η(χ) > 2.
(It can be seen that the specific values of C1 and C2 are not used in the comparison.)
1. Adan-Bante E. Products of characters and derived length // J. Algebra. – 2003. – 266. – P. 305 – 319.
2. Adan-Bante E. Products of characters with few irreducible constituents // J. Algebra. – 2007. – 311. – P. 38 – 68.
3. Adan-Bante E. Products of characters and derived length of finite solvable groups: Ph. D. Thesis. – Univ. Illinois,
Urbana, 2002.
4. Keller T. M. Orbit sizes and character degrees III // J. reine und angew. Math. – 2002. – 545. – S. 1 – 17.
Received 22.08.12,
after revision — 22.11.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 7
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| spelling | umjimathkievua-article-21952019-12-05T10:25:58Z A Note on a Bound of Adan-Bante Одне зауваження щодо границі Адан-Бантe Chen, Xiaoyou Чен, Хіаоіоу Let G be a finite solvable group and let χ be a nonlinear irreducible (complex) character of G. Also let \( \eta \) (χ) be the number of nonprincipal irreducible constituents of \( \upchi \overline{\upchi} \) , where \( \overline{\upchi} \) denotes the complex conjugate of χ. Adan-Bante proved that there exist constants C and D such that dl (G/ ker χ) ≤ C \( \eta \) (χ) +D. In the present work, we establish a bound lower than the Adan-Bante bound for \( \eta \) (χ) > 2 Нехай $G$ — скінченна розв'язна група, а $χ$ — нєлінійний незвідний (комплексний) характер групи $G$. Також нехай $η (χ)$ — число неголовних незвідних складових $χ \overline{χ}$, де $\overline{χ}$ позначає величину, комплексно спряжену до $χ$. Як доведено Адан-Банте, існують сталі $C$ та $D$ такі, що $\text{dl} (G/ \ker χ) ≤ C η(χ) + D$. В даній роботі встановлено оцінку нижчу, ніж оцінка Адан-Банте для $η (χ) > 2$. Institute of Mathematics, NAS of Ukraine 2014-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2195 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 7 (2014); 1006–1008 Український математичний журнал; Том 66 № 7 (2014); 1006–1008 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2195/1391 https://umj.imath.kiev.ua/index.php/umj/article/view/2195/1392 Copyright (c) 2014 Chen Xiaoyou |
| spellingShingle | Chen, Xiaoyou Чен, Хіаоіоу A Note on a Bound of Adan-Bante |
| title | A Note on a Bound of Adan-Bante |
| title_alt | Одне зауваження щодо границі Адан-Бантe |
| title_full | A Note on a Bound of Adan-Bante |
| title_fullStr | A Note on a Bound of Adan-Bante |
| title_full_unstemmed | A Note on a Bound of Adan-Bante |
| title_short | A Note on a Bound of Adan-Bante |
| title_sort | note on a bound of adan-bante |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2195 |
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