A short note on the noncoprime regular module problem
UDC 512.5 Considering a special configuration in which a finite group $A$ acts by automorphisms on а finite group $G$ and the semidirect product $GA$ acts on the vector space $V$ by linear transformations, we discuss the existence of a regular $A$-module in $V_{{A}}.$
Gespeichert in:
| Datum: | 2020 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6028 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512231968473088 |
|---|---|
| author | Ercan, G. Güloğlu , Ş. Ercan, G. Ercan, G. Güloğlu , Ş. |
| author_facet | Ercan, G. Güloğlu , Ş. Ercan, G. Ercan, G. Güloğlu , Ş. |
| author_sort | Ercan, G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:35Z |
| description | UDC 512.5
Considering a special configuration in which a finite group $A$ acts by automorphisms on а finite group $G$ and the semidirect product $GA$ acts on the vector space $V$ by linear transformations, we discuss the existence of a regular $A$-module in $V_{{A}}.$
|
| doi_str_mv | 10.37863/umzh.v72i11.6028 |
| first_indexed | 2026-03-24T03:25:30Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
DOI: 10.37863/umzh.v72i11.6028
UDC 512.5
G. Ercan (Middle East Techn. Univ., Ankara, Turkey),
Ş. Güloğlu (Doğuş Univ., Istanbul, Turkey)
A SHORT NOTE ON THE NONCOPRIME REGULAR MODULE PROBLEM*
КОРОТКИЙ КОМЕНТАР ЩОДО ЗАДАЧI ПРО РЕГУЛЯРНI МОДУЛI,
ЩО НЕ Є ВЗАЄМНО ПРОСТИМИ
Considering a special configuration in which a finite group A acts by automorphisms on а finite group G and the semidirect
product GA acts on the vector space V by linear transformations, we discuss the existence of a regular A-module in VA.
Розглянуто спецiальну конфiгурацiю, в якiй скiнченна група A дiє за допомогою автоморфiзмiв на скiнченну групу
G, а напiвпрямий добуток GA — на векторний простiр V за допомогою лiнiйних перетворень; обговорюється
iснування регулярного A-модуля у VA.
1. Introduction. Let A be a finite group which acts faithfully on the vector space V by linear
transformations. We say “A has a regular orbit on V ” if there is a vector v in V such that CA(v) = 1.
In this case, the A-orbit containing v is called a regular A-orbit. Furthermore, V contains the regular
A-module if a regular A-orbit happens to be linearly independent. More generally if A acts by linear
transformations on the vector space V (not necessarily faithfully), then we say that A has a regular
orbit on V or V contains the regular A-module if A/CA(V ) does the same.
While studying the structure of a finite solvable group G admitting a certain group of automor-
phisms A, we are often forced to study A-invariant chief factors V of G together with the action of
the semidirect product (G/CG(V ))A on V. It turns out to be rather efficient to know that V contains
the regular A-module or at least a regular A-orbit. Not all groups act with regular orbits although
many interesting and rich classes do, especially under the additional assumptions of coprimeness
that (| G| , | A| ) = 1 = (| V | , | GA| ). There has been extensive research about the existence of regular
orbits such as [1, 6 – 8, 11, 12] in the case of coprimeness and [2, 4, 5, 13, 14] in the noncoprime
case. All the results concerning a nilpotent A are culminating in Theorem 1.1 in [14] which can be
reformulated as follows:
Let G be a finite solvable group admitting a nilpotent group A as a group of automorphisms.
Suppose that COp(A)(G) = 1. Let V be a finite faithful kGA-module over a field k of characteristic
p not dividing the order of G. Then A has at least one regular orbit on V if A involves no wreath
product \BbbZ 2 \wr \BbbZ 2 and involves no wreath product \BbbZ r \wr \BbbZ r for r a Mersenne prime when p = 2.
In the present paper, we prove a theorem which concludes the existence of a regular module
without the coprimeness condition the prototype of which is Theorem 1.5 in [11]. This theorem was
improved as Theorem B in [5] in case where the group GA is of odd order. For the convenience
* This paper was supported by the Research Project TÜBİTAK 114F223.
c\bigcirc G. ERCAN, Ş. GÜLOĞLU, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11 1589
1590 G. ERCAN, Ş. GÜLOĞLU
of the reader, we formulate the main conclusion of Theorem 1.5 in a way suitable to emphasize the
similarities and differences between this theorem and Theorem B in [5] and our result.
Let PRA be a finite group where P is a p-group and R is an r-group for distinct primes p
and r not dividing the order of A such that P \lhd PRA and R\lhd RA. Assume that the following are
satisfied:
(a) P is an extraspecial p-group for some prime p where Z(P ) \leq Z(PRA) and CA(P ) = 1;
(b) \=R = R/R0 is of class at most two and of exponent r where R0 = CR(P ); suppose that\bigm| \bigm| CA( \=R/\Phi ( \=R)
\bigm| \bigm| is either a prime or 1;
(c) A/CA( \=R/\Phi ( \=R) has a regular orbit in its action on \=R/\Phi ( \=R);
if CA( \=R/\Phi ( \=R) \not = 1, [CA( \=R/\Phi ( \=R), P ] \not = P and p = 2, assume that
\bigm| \bigm| CA( \=R/\Phi ( \=R)
\bigm| \bigm| is not a
Fermat prime.
Let \chi be a complex PRA-character such that \chi P is faithful. Then \chi A contains the regular
A-character.
Namely we obtain the following theorem.
Theorem. Let PRA be a finite group where P is a p-group and R is an r-group for distinct
primes p and r such that P \lhd PRA and R\lhd RA. Assume that the following are satisfied:
(a) P is an extraspecial p-group for some prime p where Z(P ) \leq Z(PRA) and CA(P ) = 1;
(b) R/R0 is of class at most two and of exponent dividing r where R0 = CR(P ) and A0 =
= CA(R/R0) = 1;
(c) A = Ap \times Ar \times A\{ p,r\} \prime where its Sylow r-subgroup Ar and Sylow p-subgroup Ap are both
cyclic and A\{ p,r\} \prime acts with regular orbits on R/\Phi (R);
(d) if p = 2 then r is not a Fermat prime.
Let \chi be a complex PRA-character such that \chi P is faithful. Then \chi A contains the regular
A-character.
Notice that both p and r are allowed to divide the order of A.
All groups considered in this paper are finite and the notation and terminology are standard.
2. Existence of regular orbits. In this section, we present a result due to Dade [3] on the
existence of regular orbits which will be applied in the proof of our theorem.
Proposition. Let V be a faithful kA-module over a finite field k of characteristic p. Assume
that A = B \times C where B is a cyclic p-group and C is a p\prime -group which has a regular orbit on
every C -invariant irreducible section of V. Then A has a regular orbit on V.
Proof. Let VC = W1\oplus . . .\oplus W\ell be the decomposition of V into its C -homogeneous components.
As B and C commute, each Wi is A-invariant. Therefore it suffices to prove that A has a regular
orbit on Wi for each i = 1, . . . , \ell . To see this let wi \in Wi be such that CA(wi) = CA(Wi) for
i = 1, . . . , \ell . If v = w1 + . . .+ w\ell , then
CA(v) =
k\bigcap
i=1
CA(wi) =
k\bigcap
i=1
CA(Wi) = CA(V ) = 1.
Thus we may assume that \ell = 1, that is, VC is homogeneous. Let X be the irreducible kC -module
which appears in VC and let B = \langle \alpha \rangle . Then we have kB = k[\alpha - 1]. Set Rj = kB/
\bigl\langle
(\alpha - 1)j
\bigr\rangle
for j = 1, . . . , pn, where pn = | \alpha | . Note that Rj is an indecomposable kB-module of dimension j
for each j and these are the only indecomposable kB-modules by Theorem VII.5.3 in [9]. Then the
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
A SHORT NOTE ON THE NONCOPRIME REGULAR MODULE PROBLEM 1591
decomposition of the kA-module V into indecomposable kA-modules can be given as
V \sim = (X \otimes Rj1)\oplus . . .\oplus (X \otimes Rjm)
\sim = X \otimes
\Biggl(
m\bigoplus
i=1
Rji
\Biggr)
for some j1, . . . , jm in \{ 1, . . . , pn\} . To simplify the notation we set U =
\bigoplus m
i=1Rji . The group C
has a regular orbit on X by hypothesis, that is, there is x \in X such that CC(x) = CC(X) = 1. We
shall observe that B has a regular orbit on U : As a consequence of the faithful action of A on V, B
acts faithfully on U. Hence there is at least one indecomposable component, say Rji , on which B
acts faithfully, since B is cyclic. Let
Rji = U1 \supset U2 \supset . . . \supset Us = 0
be a B-composition series of Rji = U1. Each factor Ui/Ui+1, i = 1, . . . , s - 1, is isomorphic to the
trivial module of dimension 1. Hence s - 1 = \mathrm{d}\mathrm{i}\mathrm{m}U1 = j1 and
\biggl[
U1, \alpha , . . . , \alpha \underbrace{} \underbrace{}
j1 - times
\biggr]
= 0. It follows that
\mathrm{d}\mathrm{i}\mathrm{m}U1 = j1 \geq pn - 1 + 1, because otherwise (\alpha - 1)p
n - 1
= 0 on U1 and, hence, \alpha pn - 1
is trivial on
U1, a contradiction. Pick an element u from U1 - U2. If CB(u) \not = 1, then \alpha pn - 1
acts trivially on
u, whence the degree j1 of the minimum polynomial of \alpha on U1 is at most pn - 1 . But then pn - 1 +
+1 \leq j1 \leq pn - 1, which is impossible. This yields that CB(u) = 1 = CB(U). As a consequence, B
has a regular orbit on U. We are now ready to complete the proof of the theorem. Let a \in CA(x\otimes u).
Then a = b + c for some b \in B and c \in C. As c \in \langle a\rangle , we have (x \otimes u)c = xc \otimes u = x \otimes u
and hence xc = x. That is, c \in CC(x) = CC(X). Similarly, we observe that b \in CB(u) = CA(U).
Consequently, we have a \in CA(X \otimes U) and, hence, the equality CA(x \otimes u) = CA(X \otimes U) holds.
It follows that A has regular orbit on V, as claimed.
The proposition is proved.
Remark. The above proposition cannot be extended to Abelian Op(A) as the following example
shows: Let V be an elementary Abelian group of order p3 with a basis \{ v1, v2, v3\} and A an
elementary Abelian group of order p2 of automorphisms of V generated by \{ a1, a2\} with the action
va11 = va21 = v1, v
a1
2 = v1v2, v
a2
2 = v2, v
a1
3 = v3, v
a2
3 = v3v1. Then every A-orbit on V has length
dividing p.
3. Proof of theorem. Let (P,R, \chi ) be a counterexample with | PR| +\chi (1) minimum. We shall
proceed in a series of steps. To simplify the notation we set G = PR.
(1) \chi is irreducible.
There exists an irreducible constituent \chi 1 of \chi which does not contain Z(P ) in its kernel, that
is (\chi 1)P is faithful. Then we have \chi 1 = \chi because otherwise \chi 1 contains the regular A-character
by induction.
(2) \chi P is homogeneous and R0 = 1.
As it is well-known the irreducible characters of the extraspecial group P are uniquely determined
by their restriction Z(P ) so that \chi P = e\theta for some faithful irreducible GA-invariant character \theta of
P and some positive integer e, since Z(P ) \leq Z(GA). The coprimeness condition (| P | , | RAp\prime | ) = 1
enables us to extend \theta in a unique way to an irreducible character \theta of GAp\prime such that \mathrm{d}\mathrm{e}\mathrm{t}(\theta )(x) = 1
for each x \in RAp\prime by [10] (8.16). On the other hand \theta 1 = \theta \times 1R0 is an irreducible P\times R0-character
with R0 \leq \mathrm{K}\mathrm{e}\mathrm{r} \theta 1. We can extend \theta 1 uniquely to \theta 1 \in Irr(GAp\prime /R0) with \mathrm{d}\mathrm{e}\mathrm{t}(\theta 1)(x) = 1 for each
x \in RAp\prime /R0. The uniqueness of this extension implies R0 \leq \mathrm{K}\mathrm{e}\mathrm{r} \theta . Notice that (\theta 1)P = \theta = \theta P
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1592 G. ERCAN, Ş. GÜLOĞLU
and also that the set
\bigl\{
\varphi : \varphi \in \mathrm{I}\mathrm{r}\mathrm{r}(GAp\prime ) such that \varphi P = \theta
\bigr\}
is Ap-invariant, because \theta a = \theta for
each a \in Ap. Since \mathrm{d}\mathrm{e}\mathrm{t}(\theta
a
)(x) = 1 for each a \in Ap, the uniqueness of \theta gives \theta
a
= \theta . It follows
from [10] (Corollary 11.22) that \theta is extendible to an irreducible GA-character, say \theta . Now \theta G = \theta ,
\theta P = \theta and R0 \leq \mathrm{K}\mathrm{e}\mathrm{r} \theta = G \cap \mathrm{K}\mathrm{e}\mathrm{r} \theta . If \theta (1) < \chi 1 or R0 \not = 1, by induction applied to the group
GA/R0 over \theta we see that \theta A contains the regular A-character. Since \chi is a constituent of \theta P | GA,
there exists \beta \in \mathrm{I}\mathrm{r}\mathrm{r}(GA/P ) such that \chi = \theta \cdot \beta by [10] (6.17) and hence \chi A = \theta A \cdot \beta A. We conclude
that \chi A contains the regular A-character, while \theta A does. Therefore without loss of generality we
may assume that R0 = 1 as claimed.
(3) Theorem follows.
Theorem 1.3 in [11] applied to the group PR over \chi shows that one of the following holds:
(i) \chi R contains the regular R-character;
(ii) p = 2 and r is a Fermat prime.
By hypothesis (d) we see that (i) follows, that is \chi R contains a copy of every irreducible R-
character. On the other hand we can regard \mathrm{I}\mathrm{r}\mathrm{r}(R/\Phi (R)) as a faithful \BbbF r(A)-module which is
isomorphic to R/\Phi (R) and hence apply the proposition above to get a linear character \nu of R such
that CA(\nu ) = 1. Let V be a GA-module affording \chi and let W be the homogeneous component of
VR corresponding to \nu . Since the stabilizer in A of W is trivial, VA contains the regular A-module.
Therefore, \chi A contains the regular A-character.
The theorem is proved.
References
1. T. R. Berger, Hall – Higman type theorems, VI, J. Algebra, 51, 416 – 424 (1978).
2. W. Carlip, Regular orbits of nilpotent subgroups of solvable groups, Illinois J. Math., 38, № 2, 199 – 222 (1994).
3. E. C. Dade, Oral communication to B Huppert, Endliche Gruppen, I, Berlin (1967).
4. A. Espuelas, The existence of regular orbits, J. Algebra, 127, 259 – 268 (1989).
5. A. Espuelas, Regular orbits on symplectic modules, J. Algebra, 138, № 1, 1 – 12 (1991).
6. P. Fleischmann, Finite groups with regular orbits on vector spaces, J. Algebra, 103, № 1, 211 – 215 (1986).
7. R. Gow, On the number of characters in a p-block of a p-solvable group, J. Algebra, 65, 421 – 426 (1980).
8. B. Hargraves, The existence of regular orbits for nilpotent groups, J. Algebra, 72, 54 – 100 (1981).
9. B. Huppert, N. Blackburn, Finite Groups, II, Grundlehren Math. Wiss., Springer-Verlag, Berlin, New York, (1982).
10. I. M. Isaacs, Character theory of finite Groups, Dover Publ., Inc., New York (1994).
11. A. Turull, Fixed point free action with regular orbits, J. reine und angew. Math., 371, 67 – 91 (1986).
12. A. Turull, Supersolvable automorphism groups of solvable groups, Math. Z., 183, 47 – 73 (1983).
13. Y. Yang, Regular orbits of finite primitive solvable groups, J. Algebra, 323, 2735 – 2755 (2010).
14. Y. Yang, Regular orbits of nilpotent subgroups of solvable linear groups, J. Algebra, 325, 56 – 69 (2011).
Received 20.09.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
|
| id | umjimathkievua-article-6028 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:30Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/24/eeb4cc07cb7579ba59afafebb7318624.pdf |
| spelling | umjimathkievua-article-60282025-03-31T08:49:35Z A short note on the noncoprime regular module problem A short note on the noncoprime regular module problem Ercan, G. Güloğlu , Ş. Ercan, G. Ercan, G. Güloğlu , Ş. nilpotent group regular orbit regular module nilpotent group regular orbit regular module UDC 512.5 Considering a special configuration in which a finite group $A$ acts by automorphisms on а finite group $G$ and the semidirect product $GA$ acts on the vector space $V$ by linear transformations, we discuss the existence of a regular $A$-module in $V_{{A}}.$ УДК 512.5 Короткий коментар щодо задачі про регулярні модулі, що не є взаємно простими Розглянуто спеціальну конфігурацію, в якій скінченна група $A$ діє за допомогою автоморфізмів на скінченну групу $G,$ а напівпрямий добуток $GA$ - на векторний простір $V$ за допомогою лінійних перетворень; обговорюється існування регулярного $A$-модуля у $V_{{A}}.$ Institute of Mathematics, NAS of Ukraine 2020-11-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6028 10.37863/umzh.v72i11.6028 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 11 (2020); 1589-1592 Український математичний журнал; Том 72 № 11 (2020); 1589-1592 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6028/8786 |
| spellingShingle | Ercan, G. Güloğlu , Ş. Ercan, G. Ercan, G. Güloğlu , Ş. A short note on the noncoprime regular module problem |
| title | A short note on the noncoprime regular module problem |
| title_alt | A short note on the noncoprime regular module problem |
| title_full | A short note on the noncoprime regular module problem |
| title_fullStr | A short note on the noncoprime regular module problem |
| title_full_unstemmed | A short note on the noncoprime regular module problem |
| title_short | A short note on the noncoprime regular module problem |
| title_sort | short note on the noncoprime regular module problem |
| topic_facet | nilpotent group regular orbit regular module nilpotent group regular orbit regular module |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6028 |
| work_keys_str_mv | AT ercang ashortnoteonthenoncoprimeregularmoduleproblem AT guloglus ashortnoteonthenoncoprimeregularmoduleproblem AT ercang ashortnoteonthenoncoprimeregularmoduleproblem AT ercang ashortnoteonthenoncoprimeregularmoduleproblem AT guloglus ashortnoteonthenoncoprimeregularmoduleproblem AT ercang shortnoteonthenoncoprimeregularmoduleproblem AT guloglus shortnoteonthenoncoprimeregularmoduleproblem AT ercang shortnoteonthenoncoprimeregularmoduleproblem AT ercang shortnoteonthenoncoprimeregularmoduleproblem AT guloglus shortnoteonthenoncoprimeregularmoduleproblem |