On representations of the group of order two over local factorial rings in the weakly modular case
We study representations of the group of order 2 over local factorial rings of characteristic not 2 with residue field of characteristic 2. The main results are related to a sufficient condition of wildness of groups.
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nasplib_isofts_kiev_ua-123456789-1559132025-02-09T18:21:08Z On representations of the group of order two over local factorial rings in the weakly modular case Bondarenko, V.M. Stoika, M. We study representations of the group of order 2 over local factorial rings of characteristic not 2 with residue field of characteristic 2. The main results are related to a sufficient condition of wildness of groups. 2017 Article On representations of the group of order two over local factorial rings in the weakly modular case / V.M. Bondarenko, M. Stoika // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 7-15. — Бібліогр.: 21 назв. — англ. 1726-3255 2010 MSC:20C15, 20C20, 16G60. https://nasplib.isofts.kiev.ua/handle/123456789/155913 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
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We study representations of the group of order 2 over local factorial rings of characteristic not 2 with residue field of characteristic 2. The main results are related to a sufficient condition of wildness of groups. |
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Bondarenko, V.M. Stoika, M. |
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Bondarenko, V.M. Stoika, M. On representations of the group of order two over local factorial rings in the weakly modular case Algebra and Discrete Mathematics |
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Bondarenko, V.M. Stoika, M. |
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Bondarenko, V.M. |
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On representations of the group of order two over local factorial rings in the weakly modular case |
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On representations of the group of order two over local factorial rings in the weakly modular case |
| title_full |
On representations of the group of order two over local factorial rings in the weakly modular case |
| title_fullStr |
On representations of the group of order two over local factorial rings in the weakly modular case |
| title_full_unstemmed |
On representations of the group of order two over local factorial rings in the weakly modular case |
| title_sort |
on representations of the group of order two over local factorial rings in the weakly modular case |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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https://nasplib.isofts.kiev.ua/handle/123456789/155913 |
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On representations of the group of order two over local factorial rings in the weakly modular case / V.M. Bondarenko, M. Stoika // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 1. — С. 7-15. — Бібліогр.: 21 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT bondarenkovm onrepresentationsofthegroupofordertwooverlocalfactorialringsintheweaklymodularcase AT stoikam onrepresentationsofthegroupofordertwooverlocalfactorialringsintheweaklymodularcase |
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2025-11-29T13:29:05Z |
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2025-11-29T13:29:05Z |
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1850131568030384128 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 23 (2017). Number 1, pp. 7–15
c© Journal “Algebra and Discrete Mathematics”
On representations of the group of order two
over local factorial rings in the weakly
modular case
Vitaliy M. Bondarenko, Myroslav V. Stoika
Communicated by V. V. Kirichenko
Abstract. We study representations of the group of order 2
over local factorial rings of characteristic not 2 with residue field of
characteristic 2. The main results are related to a sufficient condition
of wildness of groups.
Introduction
A group G is called wild over an commutative ring K, if the problem of
classifying its matrix K-representations contains the problem of classifying
the pairs matrices, up to similarity, over a field k. Otherwise, G is called
tame over K. When K is a field of characteristic p (p > 0), a finite group
G is tame if and only if its every noncyclic abelian p-subgroup has order
at most 4 [1]. In particular,
1) in the classical case, when the order of G is not divisible by p,
the group G is always tame and even has, up to equivalence, only finite
number of indecomposable representations;
2) in the modular case, when the order of G is divisible by p, the
group G has only finite number of indecomposable representations if and
only if its Sylow p-subgroup Gp is cyclic; when it is not, then G is tame
if and only if p = 2 and G2/[G2, G2] ∼= (2, 2).
2010 MSC: 20C15, 20C20, 16G60.
Key words and phrases: free algebra, factorial ring, maximal ideal, perfect
representation, wild group.
8 On representations of the group of order two
For commutative rings such problem, in general case, is not solved. The
first work in this direction is due to the first author [2]. If one talks about
integral domains, then in the weakly modular case, i.e. when the order of
G is not divisible by the characteristic p of a ring K but is divisible by
the characteristic of the residue field [3], a criterion of wildness of G over
a local ring K was obtained, in particular, in the following cases:
1) K = Z ′
p
is the ring of p-adic rational numbers [4];
2) K = Rp is the ring of integers of a finite extension Fp of the field
p-adic rational numbers [5];
3) G is a p-group, K is a ring of formal power series in n variables
over a complete discrete valuation ring of characteristic 0 with residue
field of characteristic p [6].
Wildness of p-groups of order greater than p was studied in [6] for
p > 2 and in [7, 8] for p = 2. Note that the smaller order of the group,
the harder to find conditions of its wildness.
In this paper we study the case when the order of G is equal to 2.
1. Formulation of the main results
Let K be a local integral domain with maximal ideal R and residue
field k, and G be a group. A matrix representation Γ of G over the free
(associative) K-algebra Σ = K〈x, y〉 is said to be perfect if from the
equivalence of the representations Γ ⊗ T and Γ ⊗ T ′ of G over K, where
T, T ′ are matrix representations Σ over K, it follows that T and T ′ are
equivalent modulo R. Following Yu. Drozd [9, pp. 70-71] we call the group
G wild over K if it has a perfect representation over Σ1.
Recall some definitions on integral domains.
A prime element, or simply a prime, of an integral domain K is, by
definition, a non-unit (non-invertible) element c such that whenever c|ab
for some a, b ∈ K, then c|a or c|b. The element εc with ε to be a unit is
called associated to c.
A factorial ring K is an integral domain in which every non-zero non-
unit element x can be written as a product of prime elements, uniquely
up to order and unit factors. The number l(x) of the prime factors of x is
called the length of x.
1The problem of allocation of wild objects (relative to different equivalences) has
long been one of the main problems of modern representation theory. Besides classical
objects (groups, algebras, rings, etc.) there are such well-known objects as directed
graphs (quivers) and posets, both with various additional conditions (see, e.g. [10] – [13]
for graphs and [14] – [21] for posets).
V. M. Bondarenko, M. V. Stoika 9
By different prime elements of K we mean non-associated ones.
The aim of this paper is to prove the following theorem.
Theorem 1 (on six twos). Let G be the group of order 2 and K a local
factorial ring of characteristic not 2 with residue field of characteristic 2.
If K has 2 different primes and l(2) > 2, then G is wild.
Corollary 1. Let G be a (finite or infinite) group with a factor group to
be a finite 2-group, and K be as in Theorem. Then G is wild.
2. Auxiliary propositions
In this section K is a local integral domain with maximal ideal R.
Lemma 1. Let 2 = t1 t2 t (in K), where t1, t2 are different primes, t ∈ R,
and let
t2
1x + t2
2y + t1t2z = 2w (1)
for some x, y, z, w ∈ K. Then x ≡ y ≡ z ≡ 0 (mod R).
Proof. From 2 = t1 t2 t and (1),
t2(t2y + t1z − t1tw) = −t2
1x (2)
whence t2|x and therefore x ≡ 0 (mod R). Let x = t2x′. Then we have
from (2) (after reducing by t2 and elementary transformations) that
t1(z + t1x′ − tw) = −t2y
whence t1|y and t2|z + t1x′ − tw; consequently y ≡ z ≡ 0 (mod R).
Lemma 2. Let 2 = t2
1t (in K), where t1 is a prime, t ∈ R, and let
t2
1x + t2
2y + t1t2z = 2w (3)
for some x, y, z, w ∈ K and a prime t2 6= t1. Then x ≡ y ≡ z ≡ 0
(mod R).
Proof. From 2 = t2
1 t and (3),
t1(t1x + t2z − t1tw) = −t2
2y (4)
whence t1|y and therefore y ≡ 0 (mod R). Let y = t1y′. Then we have
from (4) (after reducing by t1 and elementary transformations) that
t1(x − tw) = −t2(z + t2y′)
whence t2|x − tw) and t1|z + t2y′; consequently x ≡ z ≡ 0 (mod R).
10 On representations of the group of order two
3. Proof of Theorem
Let G =
〈
g| g2 = e
〉
. It is natural to identify a matrix representations
T of Σ = K〈x, y〉 over K with the ordered pair of matrices T (x), T (y);
if these matrices are of size m × m, we say that T is of K-dimension m.
Then, for a matrix representation Γ of the group G over K (see above the
definition of a wild group) and T of K-dimension m, the matrix (Γ⊗T )(g)
is obtained from the matrix Γ(g) by change x and y on the matrices T (x)
and T (y), and a ∈ K on the scalar matrix aEm, where Em is the identity
m × m matrix.
From the conditions of the theorem it follows immediately that
1) 2 = t1t2t with t1, t2 to be different primes and t ∈ R, or
2) 2 = t2
1t with t1 to be a prime and t ∈ R.
Consider first case 1).
We prove that the representation Γ of G over Σ of the form
Γ : g →
1 0 0 t1t2 t2
1 x 0
0 1 0 t2
2 t1t2 t2
1 y
0 0 1 0 t2
2 t1t2
0 0 0 −1 0 0
0 0 0 0 −1 0
0 0 0 0 0 −1
is perfect.
Let T = (A, B) and T ′ = (A′, B′) be matrix representations of Σ over
K of a K-dimension n. Then
(Γ ⊗ T )(g) =
En 0 0 t1t2En t2
1A 0
0 En 0 t2
2En t1t2En t2
1B
0 0 En 0 t2
2En t1t2En
0 0 0 −En 0 0
0 0 0 0 −En 0
0 0 0 0 0 −En
=
=
(
E3n t2
1M(A, B) + t2
2N + t1t2E3n
0 −E3n
)
V. M. Bondarenko, M. V. Stoika 11
and
(Γ ⊗ T ′)(g) =
En 0 0 En t2
1A′ 0
0 En 0 t2
2En En t2
1B′
0 0 En 0 t2
2En En
0 0 0 −En 0 0
0 0 0 0 −En 0
0 0 0 0 0 −En
=
=
(
E3n t2
1M(A′, B′) + t2
2N + t1t2E3n
0 −E3n
)
,
where
M(A, B) =
0 A 0
0 0 B
0 0 0
, M(A′, B′) =
0 A′ 0
0 0 B′
0 0 0
and
N =
0 0 0
En 0 0
0 En 0
.
Assume that the representations Γ(A, B) and Γ(A′, B′) are equiva-
lent, i. e. there exists an invertible matrix C such that (Γ ⊗ T )(g)C =
C(Γ ⊗ T ′)(g). So we have the equality
(
E3n t2
1M(A, B) + t2
2N + t1t2E3n
0 −E3n
)(
C1 C2
C3 C4
)
=
=
(
C1 C2
C3 C4
)(
E3n t2
1M(A′, B′) + t2
2N + t1t2E3n
0 −E3n
)
,
(5)
where the partition of
C =
(
C1 C2
C3 C4
)
on blocks is compatible with those of (Γ ⊗ T )(g), (Γ ⊗ T ′)(g).
The equality (5) is equivalent to the following ones:
C1 + t2
1M(A, B)C3 + t2
2NC3 + t1t2C3 = C1,
C2 + t2
1M(A, B)C4 + t2
2NC4 + t1t2C4
= t2
1C1M(A′, B′) + t2
2C1N + t1t2C1 − C2,
−C3 = C3,
−C4 = t2
1C3M(A′, B′) + t2
2C3N + t1t2C3 − C4.
12 On representations of the group of order two
In turn, these equations are equivalent to the equations C3 = 0 and
t2
1(M(A, B)C4 −C1M(A′, B′))+t2
2(NC4 −C1N)+t1t2(C4 −C1) = −2C2.
By applying Lemma 1 to all scalar equations of the last matrix equation,
we easily see that
M(A, B)C4 ≡ C1M(A′, B′) (mod R),
NC4 ≡ C1N (mod R), C4 ≡ C1 (mod R),
or equivalently,
M(A, B)C1 ≡ C1M(A′, B′) (mod R), (6)
NC1 ≡ C1N (mod R). (7)
From C3 = 0 it follows that the block C1 of the (invertible) matrix C
is invertible. Put
C1 =
C11 C12 C13
C21 C22 C23
C31 C32 C33
and write (7) in the expanded form:
0 0 0
En 0 0
0 En 0
C11 C12 C13
C21 C22 C23
C31 C32 C33
≡
≡
C11 C12 C13
C21 C22 C23
C31 C32 C33
0 0 0
En 0 0
0 En 0
(mod R)
(the partition of C1 on blocks is compatible with those of N). From this
we have
0 0 0
C11 C12 C13
C21 C22 C23
≡
C12 C13 0
C22 C23 0
C32 C33 0
(mod R),
whence C12 ≡ C13 ≡ C23 ≡ 0 (mod R), C11 ≡ C22 ≡ C33 (mod R),
C21 ≡ C32 (mod R), and therefore
C1 ≡
C11 0 0
C21 C11 0
C31 C21 C11
(mod R) (8)
V. M. Bondarenko, M. V. Stoika 13
with C11 being invertible modulo R.
From (6) and (8),
0 A 0
0 0 B
0 0 0
C11 0 0
C21 C11 0
C31 C21 C11
≡
≡
C11 0 0
C21 C11 0
C31 C21 C11
0 A′ 0
0 0 B′
0 0 0
(mod R)
,
or equivalently
AC21 AC11 0
BC31 BC21 BC11
0 0 0
≡
0 C11A′ 0
0 C21A′ C11B′
0 C31A′ C21B′
(mod R).
From this, in particular, we have
AC11 ≡ C11A′ (mod R), BC11 ≡ C11B′ (mod R),
as required.
Now consider case 2.
In this case we take as a perfect representation Γ of G over Σ the
representation of the same form as in case 1) with t2 to be any prime
element different from t1 (it exists by the condition of the theorem). Then
the proof is analogously to that in case 1), but it is need to use Lemma 2
instead of Lemma 1.
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14 On representations of the group of order two
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V. M. Bondarenko, M. V. Stoika 15
Contact information
V. M. Bondarenko Institute of Mathematics, Tereshchenkivska 3,
01601 Kyiv, Ukraine
E-Mail(s): vitalij.bond@gmail.com
Web-page(s): http://www.imath.kiev.ua
M. V. Stoika Department of Mathematics and Informatics,
Ferenc Rakoczi II Transcarpathian Hungarian
Institute, Kossuth square 6, 90200 Beregovo,
Ukraine
E-Mail(s): stoyka−m@yahoo.com
Received by the editors: 14.02.2017.
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