Integral group ring of Rudvalis simple group
Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru . As a consequence, for this group we confirm the Kimmerle conjecture on prime graphs.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509007039430656 |
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| author | Bovdi, V. A. Konovalov, A. V. Бовді, В. А. Коновалов, А. В. |
| author_facet | Bovdi, V. A. Konovalov, A. V. Бовді, В. А. Коновалов, А. В. |
| author_sort | Bovdi, V. A. |
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| datestamp_date | 2020-03-18T19:43:07Z |
| description | Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru . As a consequence, for this group we confirm the Kimmerle conjecture on prime graphs. |
| first_indexed | 2026-03-24T02:34:15Z |
| format | Article |
| fulltext |
UDC 512.552.7
V. A. Bovdi (Univ. Debrecen, Inst. Math., Hungary),
A. B. Konovalov (Univ. St. Andrews, School Comput. Sci., Scotland)
INTEGRAL GROUP RING OF RUDVALIS SIMPLE GROUP*
IНТЕГРАЛЬНЕ ГРУПОВЕ КIЛЬЦЕ ПРОСТОЇ ГРУПИ
РУДВАЛIСА
Using the Luthar – Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit
group of the integral group ring of the Rudvalis sporadic simple group Ru . As a consequence, for this group
we confirm Kimmerle’s conjecture on prime graphs.
За допомогою методу Лутара – Пассi дослiджено класичну гiпотезу Цассенхауза для нормованої муль-
типлiкативної групи цiлочисельного групового кiльця спорадичної простої групи Рудвалiса Ru . Як
наслiдок, для цiєї групи пiдтверджено гiпотезу Кiммерле щодо графiв простих чисел.
1. Introduction, conjectures and main results. Let V (ZG) be the normalized unit
group of the integral group ring ZG of a finite group G. A long-standing conjecture of
H. Zassenhaus (ZC) says that every torsion unit u ∈ V (ZG) is conjugate within the
rational group algebra QG to an element in G (see [1]).
For finite simple groups the main tool for the investigation of the Zassenhaus
conjecture is the Luthar – Passi method, introduced in [2] to solve it for A5 and then
applied in [3] for the case of S5. Later M. Hertweck in [4] extended the Luthar – Passi
method and applied it for the investigation of the Zassenhaus conjecture for PSL(2, pn).
The Luthar – Passi method proved to be useful for groups containing non-trivial normal
subgroups as well. For some recent results we refer to [4 – 9]. Also, some related properti-
es and some weakened variations of the Zassenhaus conjecture can be found in [3, 10, 11].
First of all, we need to introduce some notation. By #(G) we denote the set of all
primes dividing the order of G. The Gruenberg – Kegel graph (or the prime graph) of G
is the graph π(G) with vertices labeled by the primes in #(G) and with an edge from
p to q if there is an element of order pq in the group G. In [12] W. Kimmerle proposed
the following weakened variation of the Zassenhaus conjecture:
(KC) If G is a finite group then π(G) = π(V (ZG)).
In particular, in the same paper W. Kimmerle verified that (KC) holds for finite
Frobenius and solvable groups. We remark that with respect to the so-called p-version
of the Zassenhaus conjecture the investigation of Frobenius groups was completed by
M. Hertweck and the first author in [13]. In [6, 14 – 18] (KC) was confirmed for the
Mathieu simple groupsM11, M12, M22, M23, M24 and the sporadic Janko simple groups
J1, J2, and J3.
Here we continue these investigations for the Rudvalis simple group Ru . Although
using the Luthar – Passi method we cannot prove the rational conjugacy for torsion units
of V (Z Ru), our main result gives a lot of information on partial augmentations of these
units. In particular, we confirm the Kimmerle’s conjecture for this group.
* The research was supported by OTKA No. K68383.
c© V. A. BOVDI, A. B. KONOVALOV, 2009
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1 3
4 V. A. BOVDI, A. B. KONOVALOV
Let G = Ru . It is well known (see [19]) that |G| = 214 · 33 · 53 · 7 · 13 · 29 and
exp(G) = 24 · 3 · 5 · 7 · 13 · 29. Let
C =
{
C1, C2a, C2b, C3a, C4a, C4b, C4c, C4d, C5a, C5b, C6a,
C7a, C8a, C8b, C8c, C10a, C10b,
C12a, C12b, C13a, C14a, C14b, C14c, C15a, C16a,
C16b, C20a, C20b, C20c, C24a, C24b, C26a, C26b, C26c, C29a, C29b
}
be the collection of all conjugacy classes of Ru, where the first index denotes the order
of the elements of this conjugacy class and C1 = {1}. Suppose u =
∑
αgg ∈ V (ZG)
has finite order k. Denote by νnt = νnt(u) = εCnt
(u) =
∑
g∈Cnt
αg the partial
augmentation of u with respect to Cnt. From the Berman – Higman Theorem (see [20]
and Ch. 5 [21, p. 102]) one knows that ν1 = α1 = 0 and∑
Cnt∈C
νnt = 1. (1)
Hence, for any character χ of G, we get that χ(u) =
∑
νntχ(hnt), where hnt is a
representative of the conjugacy class Cnt.
Our main result is the following theorem:
Theorem 1. Let G denote the Rudvalis sporadic simple group Ru . Let u be a
torsion unit of V (ZG) of order |u| and let
P(u) =
(
ν2a, ν2b, ν3a, ν4a, ν4b, ν4c, ν4d, ν5a, ν5b, ν6a, ν7a, ν8a, ν8b,
ν8c, ν10a, ν10b, ν12a, ν12b, ν13a, ν14a, ν14b, ν14c, ν15a, ν16a,
ν16b, ν20a, ν20b, ν20c, ν24a, ν24b, ν26a, ν26b, ν26c, ν29a, ν29b
)
∈ Z35
be the tuple of partial augmentations of u. The following properties hold.
(i) If |u| 6∈ {28, 30, 40, 48, 52, 56, 60, 80, 104, 112, 120, 208, 240}, then |u|
coincides with the order of some element g ∈ G. Equivalently, there is no elements of
orders 21, 35, 39, 58, 65, 87, 91, 145, 203 and 377 in V (ZG).
(ii) If |u| ∈ {3, 7, 13}, then u is rationally conjugate to some g ∈ G.
(iii) If |u| = 2, the tuple of the partial augmentations of u belongs to the set{
P(u) | ν2a + ν2b = 1, −10 ≤ ν2a ≤ 11, νkx = 0, kx 6∈ {2a, 2b}
}
.
(iv) If |u| = 5, the tuple of the partial augmentations of u belongs to the set{
P(u) | ν5a + ν5b = 1, −1 ≤ ν5a ≤ 6, νkx = 0, kx 6∈ {5a, 5b}
}
.
(v) If |u| = 29, the tuple of the partial augmentations of u belongs to the set{
P(u) | ν29a + ν29b = 1, −4 ≤ ν29a ≤ 5, νkx = 0, kx 6∈ {29a, 29b}
}
.
As an immediate consequence of part (i) of the Theorem 1 we obtain the following
corollary:
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
INTEGRAL GROUP RING OF RUDVALIS SIMPLE GROUP 5
Corollary 1. If G = Ru then π(G) = π(V (ZG)).
2. Preliminaries. The following result is a reformulation of the Zassenhaus
conjecture in terms of vanishing of partial augmentations of torsion units.
Proposition 1 (see [2] and Theorem 2.5 in [22]). Let u ∈ V (ZG) be of order k.
Then u is conjugate in QG to an element g ∈ G if and only if for each d dividing k
there is precisely one conjugacy class C with partial augmentation εC(ud) 6= 0.
The next result now yield that several partial augmentations are zero.
Proposition 2 (see [7], Proposition 3.1; [4], Proposition 2.2). Let G be a finite
group and let u be a torsion unit in V (ZG). If x is an element of G whose p-part,
for some prime p, has order strictly greater than the order of the p-part of u, then
εx(u) = 0.
The key restriction on partial augmentations is given by the following result that is
the cornerstone of the Luthar – Passi method.
Proposition 3 (see [2, 4]). Let either p = 0 or p a prime divisor of |G|. Suppose
that u ∈ V (ZG) has finite order k and assume k and p are coprime in case p 6= 0. If
z is a complex primitive k-th root of unity and χ is either a classical character or a
p-Brauer character of G, then for every integer l the number
µl(u, χ, p) =
1
k
∑
d|k
TrQ(zd)/Q{χ(ud)z−dl} (2)
is a non-negative integer.
Note that if p = 0, we will use the notation µl(u, χ, ∗) for µl(u, χ, 0).
Finally, we shall use the well-known bound for orders of torsion units.
Proposition 4 (see [23]). The order of a torsion element u ∈ V (ZG) is a divisor
of the exponent of G.
3. Proof of Theorem 1. Throughout this section we denote the group Ru by
G. The character table of G, as well as the p-Brauer character tables, which will be
denoted by BCT(p) where p ∈ {2, 3, 5, 7, 13, 29}, can be found using the computational
algebra system GAP [24], which derives these data from [25, 26]. For the characters
and conjugacy classes we will use throughout the paper the same notation, indexation
inclusive, as used in the GAP Character Table Library.
Since the group G possesses elements of orders 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15,
16, 20, 24, 26 and 29, first of all we investigate units of some these orders (except the
units of orders 4, 6, 8, 10, 12, 14, 15, 16, 20, 24 and 26). After this, by Proposition 4,
the order of each torsion unit divides the exponent of G, so to prove the Kimmerle’s
conjecture, it remains to consider units of orders 21, 35, 39, 58, 65, 87, 91, 145, 203
and 377. We will prove that no units of all these orders do appear in V (ZG).
Now we consider each case separately.
Let u be an involution. By (1) and Proposition 2 we have that ν2a + ν2b = 1. Put
t1 = 3ν2a − 7ν2b and t2 = 11ν2a − 7ν2b. Applying Proposition 3 we get the following
system
µ1(u, χ2, ∗) =
1
2
(2t1 + 378) ≥ 0, µ0(u, χ2, ∗) =
1
2
(−2t1 + 378) ≥ 0,
µ0(u, χ4, ∗) =
1
2
(2t2 + 406) ≥ 0, µ1(u, χ4, ∗) =
1
2
(−2t2 + 406) ≥ 0.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
6 V. A. BOVDI, A. B. KONOVALOV
From these restrictions and the requirement that all µi(u, χj , ∗) must be non-negative
integers we get 22 pairs (ν2a, ν2b) listed in part (iii) of the Theorem 1.
Note that using our implementation of the Luthar – Passi method, which we intended
to make available in the GAP package LAGUNA [27], we computed inequalities from
Proposition 3 for every irreducible character from ordinary and Brauer character tables,
and for every 0 ≤ l ≤ |u| − 1, but the only inequalities that really matter are those ones
listed above. The same remark applies for all other orders of torsion units considered in
the paper.
Let u be a unit of order either 3, 7 or 13. Using Proposition 2 we obtain that all partial
augmentations except one are zero. Thus by Proposition 1 part (ii) of the Theorem 1 is
proved.
Let u be a unit of order 5. By (1) and Proposition 2 we get ν5a + ν5b = 1. Put
t1 = 6ν5a + ν5b and t2 = 3ν5a − 2ν5b. By (2) we obtain the system of inequalities
µ0(u, χ4, ∗) =
1
5
(4t1 + 406) ≥ 0, µ1(u, χ4, ∗) =
1
5
(−t1 + 406) ≥ 0,
µ0(u, χ2, 2) =
1
5
(4t2 + 28) ≥ 0, µ1(u, χ2, 2) =
1
5
(−t2 + 28) ≥ 0.
Again, using the condition for µi(u, χj , p) to be non-negative integers, we obtain eight
pairs (ν5a, ν5b) listed in part (iv) of the Theorem 1.
Let u be a unit of order 29. By (1) and Proposition 2 we have that ν29a + ν29b = 1.
Put t1 = 15ν29a − 14ν29b. Then using (2) we obtain the system of inequalities
µ1(u, χ6, 2) =
1
29
(t1 + 8192) ≥ 0, µ2(u, χ7, 5) =
1
29
(−t1 + 2219) ≥ 0,
µ1(u, χ2, 5) =
1
29
(12ν29a − 17ν29b + 133) ≥ 0,
µ2(u, χ2, 5) =
1
29
(−17ν29a + 12ν29b + 133) ≥ 0.
Now applying the condition for µi(u, χj , p) to be non-negative integers we obtain ten
pairs (ν29a, ν29b) listed in part (v) of the Theorem 1.
Now it remains to prove part (i) of the Theorem 1.
Let u be a unit of order 21. By (1) and Proposition 2 we obtain that ν3a + ν7a = 1.
By (2) we obtain the system of inequalities
µ1(u, χ4, ∗) =
1
21
(ν3a + 405) ≥ 0,
µ0(u, χ2, 2) =
1
21
(12ν3a + 30) ≥ 0,
µ7(u, χ2, 2) =
1
21
(−6ν3a + 27) ≥ 0,
which has no integer solutions such that all µi(u, χj , p) are non-negative integers.
Let u be a unit of order 35. By (1) and Proposition 2 we get ν5a + ν7a + ν7b = 1.
Put t1 = ν5a + ν5b. Since |u7| = 5, for any character χ of G we need to consider eight
cases defined by part (iv) of the Theorem 1. Using (2), in all of these cases we get the
same system of inequalities
µ0(u, χ2, ∗) =
1
35
(72t1 + 390) ≥ 0,
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
INTEGRAL GROUP RING OF RUDVALIS SIMPLE GROUP 7
µ0(u, χ4, 2) =
1
35
(−96t1 + 1230) ≥ 0,
which has no integer solutions such that all µi(u, χj , p) are non-negative integers.
Let u be a unit of order 39. By (1) and Proposition 2 we have that ν3a + ν13a = 1.
By (2) we obtain that
µ0(u, χ5, ∗) =
1
39
(72ν13a + 819) ≥ 0,
µ13(u, χ5, ∗) =
1
39
(−36ν13a + 819) ≥ 0,
µ1(u, χ2, ∗) =
1
39
(ν13a + 377) ≥ 0,
µ13(u, χ2, 2) =
1
39
(−12ν3a − 24ν13a + 51) ≥ 0.
From the first two inequalities we obtain that ν13a ∈ {0, 13}, and now the last two
inequalities lead us to a contradiction.
Let u be a unit of order 58. By (1) and Proposition 2 we have that
ν2a + ν2b + ν29a + ν29b = 1.
Put t1 = 6ν2a−14ν2b−ν29a−ν29b, t2 = 11ν2a−7ν2b and t3 = 64ν2b+14ν29a−15ν29b.
Since |u2| = 29 and |u29| = 2, according to parts (iii) and (v) of the Theorem 1 we need
to consider 220 cases, which we can group in the following way. First, let
χ(u29) ∈
{
χ(2a), −5χ(2a) + 6χ(2b), −10χ(2a) + 11χ(2b),
−2χ(2a) + 3χ(2b), −8χ(2a) + 9χ(2b), 6χ(2a)− 5χ(2b),
3χ(2a)− 2χ(2b), 9χ(2a)− 8χ(2b), 4χ(2a)− 3χ(2b)
}
.
Then by (2) we obtain the system of inequalities
µ0(u, χ2, ∗) =
1
58
(−28t1 + α) ≥ 0,
µ29(u, χ2, ∗) =
1
58
(28t1 + β) ≥ 0,
µ1(u, χ2, ∗) =
1
58
(−t1 + γ) ≥ 0,
where
(α, β, γ) =
(400, 412, 383), if χ(u29) = χ(2a);
(520, 292, 263), if χ(u29) = −5χ(2a) + 6χ(2b);
(620, 192, 163), if χ(u29) = −10χ(2a) + 11χ(2b);
(460, 352, 323), if χ(u29) = −2χ(2a) + 3χ(2b);
(580, 232, 203), if χ(u29) = −8χ(2a) + 9χ(2b);
(300, 512, 483), if χ(u29) = 6χ(2a)− 5χ(2b);
(360, 452, 423), if χ(u29) = 3χ(2a)− 2χ(2b);
(240, 572, 543), if χ(u29) = 9χ(2a)− 8χ(2b);
(340, 472, 443), if χ(u29) = 4χ(2a)− 3χ(2b),
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
8 V. A. BOVDI, A. B. KONOVALOV
which has no integral solution such that all µi(u, χj , p) are non-negative integers.
In the remaining cases we consider the following system obtained by (2):
µ0(u, χ2, ∗) =
1
58
(−28t1 + α1) ≥ 0,
µ29(u, χ2, ∗) =
1
58
(28t1 + α2) ≥ 0,
µ0(u, χ4, ∗) =
1
58
(56t2 + α3) ≥ 0,
µ29(u, χ4, ∗) =
1
58
(−56t2 + α4) ≥ 0,
µ1(u, χ34, ∗) =
1
58
(−t3 + β1) ≥ 0,
µ4(u, χ34, ∗) =
1
58
(t3 + β2) ≥ 0,
where the tuple of coefficients (α1, α2, α3, α4) depends only of the value of
χ(u29):
(α1, α2, α3, α4) =
(420, 392, 392, 420), if χ(u29) = χ(2b);
(320, 492, 572, 240), if χ(u29) = 5χ(2a)− 4χ(2b);
(600, 212, 68, 744), if χ(u29) = −9χ(2a) + 10χ(2b);
(540, 272, 176, 636), if χ(u29) = −6χ(2a) + 7χ(2b);
(380, 432, 464, 348), if χ(u29) = 2χ(2a)− χ(2b);
(260, 552, 680, 132), if χ(u29) = 8χ(2a)− 7χ(2b);
(480, 332, 284, 528), if χ(u29) = −3χ(2a) + 4χ(2b);
(500, 312, 248, 564), if χ(u29) = −4χ(2a) + 5χ(2b);
(200, 612, 24, 24), if χ(u29) = 11χ(2a)− 10χ(2b);
(220, 592, 752, 60), if χ(u29) = 10χ(2a)− 9χ(2b);
(280, 532, 644, 168), if χ(u29) = 7χ(2a)− 6χ(2b);
(440, 372, 356, 456), if χ(u29) = −χ(2a) + 2χ(2b);
(560, 252, 140, 672), if χ(u29) = −7χ(2a) + 8χ(2b),
while the pair (β1, β2) depends both on χ(u29) and χ(u2):
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
INTEGRAL GROUP RING OF RUDVALIS SIMPLE GROUP 9
χ(29a) χ(29b)
χ(2b) 110641, 110513 110670, 110542
5χ(2a)− 4χ(2b) 110321, 110833 110350, 110862
−9χ(2a) + 10χ(2b) 111217, 109937 111246, 109966
−6χ(2a) + 7χ(2b) 111025, 110129 111054, 110158
2χ(2a)− χ(2b) 110513, 110641 110542, 110670
8χ(2a)− 7χ(2b) 110129, 111025 110158, 111054
−3χ(2a) + 4χ(2b) 110833, 110321 110862, 110350
−4χ(2a) + 5χ(2b) 110897, 110257 110926, 110286
11χ(2a)− 10χ(2b) 109937, 111217 109966, 111246
10χ(2a)− 9χ(2b) 110001, 111153 110030, 111182
7χ(2a)− 6χ(2b) 110193, 110961 110222, 110990
−χ(2a) + 2χ(2b) 110705, 110449 110734, 110478
−7χ(2a) + 8χ(2b) 111089, 110065 111118, 110094
5χ(29a)− 4χ(29b) −2χ(29a) + 3χ(29b)
χ(2b) 110525, 110397 110728, 110600
5χ(2a)− 4χ(2b) 110205, 110717 110408, 110920
−9χ(2a) + 10χ(2b) 111101, 109821 111304, 110024
−6χ(2a) + 7χ(2b) 110909, 110013 111112, 110216
2χ(2a)− χ(2b) 110397, 110525 110600, 110728
8χ(2a)− 7χ(2b) 110013, 110909 110216, 111112
−3χ(2a) + 4χ(2b) 110717, 110205 110920, 110408
−4χ(2a) + 5χ(2b) 110781, 110141 110984, 110344
11χ(2a)− 10χ(2b) 109821, 111101 110024, 111304
10χ(2a)− 9χ(2b) 109885, 111037 110088, 111240
7χ(2a)− 6χ(2b) 110077, 110845 110280, 111048
−χ(2a) + 2χ(2b) 110589, 110333 110792, 110536
−7χ(2a) + 8χ(2b) 110973, 109949 111176, 110152
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
10 V. A. BOVDI, A. B. KONOVALOV
2χ(29a)−
−χ(29b)
−3χ(29a)+
+4χ(29b)
−4χ(29a)+
+5χ(29b)
χ(2b) 110612, 110484 110757, 110629 110786, 110658
5χ(2a)− 4χ(2b) 110292, 110804 110437, 110949 110466, 110978
−9χ(2a) + 10χ(2b) 111188, 109908 111333, 110053 111362, 110082
−6χ(2a) + 7χ(2b) 110996, 110100 111141, 110245 111170, 110274
2χ(2a)− χ(2b) 110484, 110612 110629, 110757 110658, 110786
8χ(2a)− 7χ(2b) 110100, 110996 110245, 111141 110274, 111170
−3χ(2a) + 4χ(2b) 110804, 110292 110949, 110437 110978, 110466
−4χ(2a) + 5χ(2b) 110868, 110228 111013, 110373 111042, 110402
11χ(2a)− 10χ(2b) 109908, 111188 110053, 111333 110082, 111362
10χ(2a)− 9χ(2b) 109972, 111124 110117, 111269 110146, 111298
7χ(2a)− 6χ(2b) 110164, 110932 110309, 111077 110338, 111106
−χ(2a) + 2χ(2b) 110676, 110420 110821, 110565 110850, 110594
−7χ(2a) + 8χ(2b) 111060, 110036 111205, 110181 111234, 110210
3χ(29a)−
−2χ(29b)
−χ(29a)+
+2χ(29b)
4χ(29a)−
−3χ(29b)
χ(2b) 110583, 110455 110699, 110571 110554, 110426
5χ(2a)− 4χ(2b) 110263, 110775 110379, 110891 110234, 110746
−9χ(2a) + 10χ(2b) 111159, 109879 111275, 109995 111130, 109850
−6χ(2a) + 7χ(2b) 110967, 110071 111083, 110187 110938, 110042
2χ(2a)− χ(2b) 110455, 110583 110571, 110699 110426, 110554
8χ(2a)− 7χ(2b) 110071, 110967 110187, 11108 110042, 110938
−3χ(2a) + 4χ(2b) 110775, 110263 110891, 110379 110746, 110234
−4χ(2a) + 5χ(2b) 110839, 110199 110955, 110315 110810, 110170
11χ(2a)− 10χ(2b) 109879, 111159 109995, 111275 109850, 111130
10χ(2a)− 9χ(2b) 109943, 111095 110059, 111211 109914, 111066
7χ(2a)− 6χ(2b) 110135, 110903 110251, 111019 110106, 110874
−χ(2a) + 2χ(2b) 110647, 110391 110763, 110507 110618, 110362
−7χ(2a) + 8χ(2b) 111031, 110007 111147, 110123 111002, 109978
Additionally, when χ(u29) ∈ {χ(2b), 7χ(2a)−6χ(2b),−7χ(2a)+8χ(2b)}, we need
to consider one more inequality
µ1(u, χ2, ∗) =
1
58
(−6ν2a + 14ν2b + ν29a + ν29b + γ) ≥ 0,
where
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
INTEGRAL GROUP RING OF RUDVALIS SIMPLE GROUP 11
γ =
363, if χ(u29) = χ(2b);
503, if χ(u29) = 7χ(2a)− 6χ(2b);
223, if χ(u29) = −7χ(2a) + 8χ(2b).
All systems of inequalities, constructed as described above, have no integer solutions
such that all µi(u, χj , p) are non-negative integers.
Let u be a unit of order 65. By (1) and Proposition 2 we have that
ν5a + ν5b + ν13a = 1.
Since |u13| = 5, we need to consider eight cases listed in part (iv) of the Theorem 1.
Put t1 = 3ν5a + 3ν5b + ν13a and t2 = 6ν5a + ν5b + 3ν13a. Then using (2) we obtain
µ0(u, χ2, ∗) =
1
65
(48t1 + 402) ≥ 0,
µ13(u, χ2, ∗) =
1
65
(−12t1 + 387) ≥ 0,
µ0(u, χ4, ∗) =
1
65
(48t2 + α) ≥ 0,
µ13(u, χ4, ∗) =
1
65
(−12t2 + β) ≥ 0,
where
(α, β) =
(466, 436), if χ(u13) = χ(5a);
(446, 441), if χ(u13) = χ(5b);
(546, 416), if χ(u13) = 5χ(5a)− 4χ(5b);
(486, 431), if χ(u13) = 2χ(5a)− χ(5b);
(566, 411), if χ(u13) = 6χ(5a)− 5χ(5b);
(506, 426), if χ(u13) = 3χ(5a)− 2χ(5b);
(426, 446), if χ(u13) = −χ(5a) + 2χ(5b);
(526, 421), if χ(u13) = 4χ(5a)− 3χ(5b).
In all cases we have no solutions such that all µi(u, χi, p) are non-negative integers.
Let u be a unit of order 87. By (1) and Proposition 2 we have that
ν3a + ν29a + ν29b = 1.
Since |u3| = 29, according to part (v) of the Theorem 1 we need to consider ten cases.
Put t1 = ν29a + ν29b. In all of these cases by (2) we get the system
µ0(u, χ2, ∗) =
1
87
(56t1 + 406) ≥ 0,
µ29(u, χ2, ∗) =
1
87
(−28t1 + 406) ≥ 0,
that lead us to a contradiction.
Let u be a unit of order 91. By (1) and Proposition 2 we get ν7a + ν13a = 1. Now
using (2) we obtain non-compatible inequalities
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 1
12 V. A. BOVDI, A. B. KONOVALOV
µ0(u, χ2, 2) =
1
91
(144ν13a + 52) ≥ 0,
µ7(u, χ2, 2) =
1
91
(−12ν13a + 26) ≥ 0.
Let u be a unit of order 145. By (1) and Proposition 2 we have that
ν5a + ν5b + ν29a + ν29b = 1.
Put t1 = 3ν5a + 3ν5b + ν29a + ν29b. Since |u29| = 5 and |u5| = 29, for any character
χ of G we need to consider 80 cases defined by parts (iv) and (v) of the Theorem 1.
Luckily, in every case by (2) we obtain the same pair of incompatible inequalities
µ0(u, χ2, ∗) =
1
145
(112t1 + 418) ≥ 0,
µ29(u, χ2, ∗) =
1
145
(−28t1 + 403) ≥ 0.
Let u be a unit of order 203. By (1) and Proposition 2 we have that
ν7a + ν29a + ν29b = 1.
Since |u7| = 29, according to part (v) of the Theorem 1 we need to consider ten cases.
Put t1 = ν29a +ν29b, and then using (2) in each case we obtain a non-compatible system
of inequalities
µ29(u, χ2, 2) =
1
203
(28t1) ≥ 0,
µ0(u, χ2, 2) =
1
203
(−168t1) ≥ 0,
µ1(u, χ2, ∗) =
1
203
(t1 + 377) ≥ 0.
Let u be a unit of order 377. By (1) and Proposition 2 we have that
ν13a + ν29a + ν29b = 1.
Since |u13| = 29, we need to consider ten cases defined by part (v) of the Theorem 1.
In each case by (2) we obtain the following system of inequalities
µ0(u, χ4, ∗) =
1
377
(1008ν13a + 442) ≥ 0,
µ29(u, χ4, ∗) =
1
377
(−84ν13a + 403) ≥ 0.
which have no solution such that all µi(u, χj , ∗) are non-negative integers.
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Received 21.11.07
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|
| id | umjimathkievua-article-2996 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:34:15Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e6/d4840f8850367af477c7aa3b26910fe6.pdf |
| spelling | umjimathkievua-article-29962020-03-18T19:43:07Z Integral group ring of Rudvalis simple group Інтегральне групове кільце простої групи Рудваліса Bovdi, V. A. Konovalov, A. V. Бовді, В. А. Коновалов, А. В. Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru . As a consequence, for this group we confirm the Kimmerle conjecture on prime graphs. За допомогою методу Лутара - Пассі досліджено класичну гіпотезу Цассенхауза для нормованої мультиплікативної групи цілочисельного групового кільця спорадичної простої групи Рудваліса Ru . Як наслідок, для цієї групи підтверджено гіпотезу Кіммерле щодо графів простих чисел. Institute of Mathematics, NAS of Ukraine 2009-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2996 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 1 (2009); 3-13 Український математичний журнал; Том 61 № 1 (2009); 3-13 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2996/2741 https://umj.imath.kiev.ua/index.php/umj/article/view/2996/2742 Copyright (c) 2009 Bovdi V. A.; Konovalov A. V. |
| spellingShingle | Bovdi, V. A. Konovalov, A. V. Бовді, В. А. Коновалов, А. В. Integral group ring of Rudvalis simple group |
| title | Integral group ring of Rudvalis simple group |
| title_alt | Інтегральне групове кільце простої групи Рудваліса |
| title_full | Integral group ring of Rudvalis simple group |
| title_fullStr | Integral group ring of Rudvalis simple group |
| title_full_unstemmed | Integral group ring of Rudvalis simple group |
| title_short | Integral group ring of Rudvalis simple group |
| title_sort | integral group ring of rudvalis simple group |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2996 |
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