A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$
UDC 512.5 Let $R$ be a ring, and $ SL(2,R)$ be the special linear group of $2\times2$ matrices with determinant $1$ over $R$. We obtain the Wedderburn decomposition of$\dfrac{\mathbb{F}_q SL(2,\mathbb{Z}_3)}{J(\mathbb{F}_q SL(2,\mathbb{Z}_3))}$ and show that $ 1+J(\mathbb{F}_q SL(2,\mathbb{Z}_3))$ i...
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2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507051273224192 |
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| author | Maheshwari , S. Sharma , R. K. MAHESHWARI, SWATI SHARMA, R. K. Maheshwari , S. Sharma , R. K. |
| author_facet | Maheshwari , S. Sharma , R. K. MAHESHWARI, SWATI SHARMA, R. K. Maheshwari , S. Sharma , R. K. |
| author_sort | Maheshwari , S. |
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| datestamp_date | 2025-03-31T08:47:35Z |
| description | UDC 512.5
Let $R$ be a ring, and $ SL(2,R)$ be the special linear group of $2\times2$ matrices with determinant $1$ over $R$. We obtain the Wedderburn decomposition of$\dfrac{\mathbb{F}_q SL(2,\mathbb{Z}_3)}{J(\mathbb{F}_q SL(2,\mathbb{Z}_3))}$ and show that $ 1+J(\mathbb{F}_q SL(2,\mathbb{Z}_3))$ is a non-Abelian group, where $\mathbb{F}_q$ is a finite field with $q = p^k$ elements of characteristic $2$ and $3.$
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| first_indexed | 2026-03-24T02:03:10Z |
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DOI: 10.37863/umzh.v73i8.588
UDC 512.5
S. Maheshwari (KIET Group of Institutions Delhi-NCR, Ghaziabad, India),
R. K. Sharma (Indian Inst. Technology Delhi, India)
A NOTE ON UNITS IN \BbbF \bfitq \bfitS \bfitL (\bftwo ,\BbbZ \bfthree )
ПОВIДОМЛЕННЯ ПРО ОДИНИЦI У \BbbF \bfitq \bfitS \bfitL (\bftwo ,\BbbZ \bfthree )
Let R be a ring, and SL(2, R) be the special linear group of 2 \times 2 matrices with determinant 1 over R. We obtain the
Wedderburn decomposition of
\BbbF qSL(2,\BbbZ 3)
J(\BbbF qSL(2,\BbbZ 3))
and show that 1 + J(\BbbF qSL(2,\BbbZ 3)) is a non-Abelian group, where \BbbF q is
a finite field with q = pk elements of characteristic 2 and 3.
Нехай R — кiльце, а SL(2, R) — спецiальна лiнiйна група (2 \times 2)-матриць над R з детермiнантом 1. Отримано
декомпозицiю Веддербурна для
\BbbF qSL(2,\BbbZ 3)
J(\BbbF qSL(2,\BbbZ 3))
i показано, що 1+J(\BbbF qSL(2,\BbbZ 3)) є неабелевою групою, де \BbbF q —
скiнченне поле з q = pk елементами та характеристикою 2 або 3.
1. Introduction. Let \BbbF G be a group algebra of a group G over a field \BbbF and \scrU (\BbbF G) denotes the
unit group of \BbbF G. It is a classical problem to study units and their properties in group ring theory.
The case, when G is a finite Abelian group and characteristic of \BbbF does not divide order of G, the
structure of \BbbF G is studied by Perlis and Walker in [18]. If characteristic of \BbbF divides order of G,
the structure of \scrU (\BbbF G) is studied by Makhijani [12, p. 10 – 12]. Hurley introduced a correspondence
between group ring and certain ring of matrices [6]. As an application of units of a group ring,
Hurley gave a method to construct convolutional codes from units in group rings [7].
Many authors have found the unit group of group algebra \BbbF qG, where G is a finite non-Abelian
group and \BbbF q denotes a finite field with q = pk elements. Monaghan [17] has found \scrU (\BbbF qG), for
some non-Abelian groups G of order 24 over a field of characteristic 3. In this paper, we have
obtained the Wedderburn decomposition of \BbbF qG/J(\BbbF qG) for G = SL(2,\BbbZ 3) over a finite field of
characteristic 2 and 3. When characteristic of \BbbF q does not divide order of G, then the structure of
\scrU (\BbbF qG) for G = SL(2,\BbbZ 3) has been obtained by Maheshwari et al. [11]. Here we are providing
some literature survey for the same. For dihedral group, the structure of the unit group \scrU (\BbbF qG) has
been discussed in [1, 5, 13, 14]. Gildea et al. [4] and Sharma et al. [19] have given the structure of
the unit group \scrU (\BbbF qG), where G is alternating group A4. The unit group of group algebras of some
non-Abelian groups of small orders have been studied in [9, 20 – 22].
2. Preliminaries. We are summarizing some results that provide useful information about the
decomposition of A/J(A), where A = \BbbF qG and J(A) be its Jacobson radical. For basic definitions
and results, see [16]. We briefly introduce some definitions and notations those will be needed
subsequently.
Let G be a finite group and \BbbF q be a finite field with characteristic p. We have some definitions
due to Ferraz.
Definition 2.1. An element g \in G is said to be p-regular if p does not divide order of g. Let l
be the l.c.m. of the orders of the p-regular elements of G, \eta be a primitive lth root of unity over
\BbbF q. Then TG,\BbbF q be the multiplicative group consisting of those integers t, taken modulo s, for which
\zeta \mapsto \rightarrow \eta t defines an automorphism of \BbbF q(\eta ) over \BbbF q.
c\bigcirc S. MAHESHWARI, R. K. SHARMA, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8 1147
1148 S. MAHESHWARI, R. K. SHARMA
Note that if q is a power of a prime such that (q, l) = 1 and d = ordl (q) is the multiplicative
order of q modulo l, then
TG,Fq =
\Bigl\{
1, q, . . . , qd - 1
\Bigr\}
mod s
and Fq(\zeta ) \sim = Fqd follow using [10] (Theorem 2.21).
Definition 2.2. If g \in G is a p-regular element, then the sum of all conjugates of g \in G is
denoted by \gamma g and the cyclotomic \BbbF q -class of g is defined to be the set
S\BbbF q(\gamma g) =
\bigl\{
\gamma gt | t \in TG,\BbbF q
\bigr\}
.
Proposition 2.1 ([3], Theorem 1.2). The number of simple components of \BbbF qG/J(\BbbF qG) is equal
to the number of cyclotomic \BbbF q -classes in G.
Theorem 2.1 ([3], Theorem 1.3). Suppose that Gal(\BbbF q(\zeta )/\BbbF q) is cyclic. Let w be the number
of cyclotomic \BbbF q -classes in G. If K1,K2, . . . ,Kw are the simple components of Z(\BbbF qG/J(\BbbF qG))
and S1, S2, . . . , Sw are the cyclotomic \BbbF q -classes of G, then with a suitable reordering of indices
| Si | = [Ki : \BbbF q].
Proposition 2.2 ([8, p. 31], Proposition 6.24). Let f : R \rightarrow S be a surjective homomorphism of
rings. Then f(J(R)) \subseteq J(S) with equality if \mathrm{k}\mathrm{e}\mathrm{r}, f \subseteq J(R).
Proposition 2.3 ([8, p. 108], Proposition 1.7). Let G be a finite group and \BbbF q be a finite field.
Then G \cap (1 + J(\BbbF qG)) = OpG, where OpG denotes the maximal normal p-subgroup of G.
Theorem 2.2 ([13], Lemma 3.2). Let \BbbF be a perfect field, G be a finite group and J(\BbbF G) be
the Jacobson radical of \BbbF G. Then
\scrU (\BbbF G) = (1 + J(\BbbF G))\rtimes \scrU (\BbbF G/J(\BbbF G)).
Lemma 2.1 ([15], Lemma 3.1). Let \frakB 1, \frakB 2 be two finite dimensional F -algebras such that
\frakB 2 is semisimple. If f : \frakB 1 \rightarrow \frakB 2 is an onto homomorphism of F -algebras, then there exists a
semisimple F -algebra \ell such that
\frakB 1/J(\frakB 1) \sim = \ell \oplus \frakB 2.
Theorem 2.3 ([2, p. 146], Theorem 7.9(i)). Let q be a power of a prime. If E is a finite field
extension of \BbbF q, then
E \otimes \BbbF q (\BbbF qG/J(\BbbF qG)) \sim =
\bigl(
E \otimes \BbbF q \BbbF qG
\bigr)
/
\bigl(
E \otimes \BbbF q J(\BbbF qG)
\bigr)
,
J
\bigl(
E \otimes \BbbF q \BbbF qG
\bigr)
= E \otimes \BbbF q J(\BbbF qG).
Theorem 2.4 ([8, p. 110], Proposition 1.9). Let N be a normal subgroup of G such that G/N
is p-solvable. If | G/N | = npa, where (n, p) = 1, then
J(\BbbF qG)p
a \subseteq \BbbF qGJ(\BbbF qN) \subseteq J(\BbbF qG).
In particular, if G is p-solvable of order npa, where (n, p) = 1, then J(\BbbF qG)p
a
= 0.
Corollary 2.1 ([15], Corollary 3.3). Let q be a power of prime. Then, for any k,m \in \BbbN ,
\BbbF qk \otimes \BbbF q \BbbF qm
\sim =
\Bigl(
\BbbF
q
lm,k
\Bigr) (m,k)
as \BbbF qk -algebras, where lm,k = l.c.m.(m, k) and (m, k) = g.c.d.(m, k).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
A NOTE ON UNITS IN \BbbF qSL(2,\BbbZ 3) 1149
Throughout this paper, \BbbF q is a field of characteristic p, where q is a power of positive prime
integer. The conjugacy class of g \in G is denoted by [g].
We can see that G has 7 conjugacy classes as follows:
Representative Elements in the class Order of element
[x] x, (yx)4, (xy)4, y - 1xy 3
[x - 1] x - 1, (yx)2, (xy)2, xyx 3
[y] y, y - 1, x2yx, xyx2, xy - 1x2, x2y - 1x 4
[y2] y2 2
[xy] xy, yx, x2yx2, xy2 6
[(xy) - 1] (xy) - 1, x2y - 1, xy - 1x, x2y2 6
Theorem 2.5. Let p = 2, q = pk and G = SL(2,\BbbZ 3). Then
\BbbF qG
J(\BbbF qG)
\sim = \BbbF q \oplus \BbbF q2 .
Proof. Suppose k is odd. Hence there exists an element of order 3, say \eta \in \BbbF q2 \setminus \BbbF q. We define
the \BbbF q -algebra homomorphism
\theta : \BbbF qG - \rightarrow \BbbF q \oplus \BbbF q2
by the assignment
x \mapsto \rightarrow (1, \eta ), y \mapsto \rightarrow (1, 1).
By using Table 1, we see that \theta is onto.
Table 1. Ontoness of \theta
Basis element Pre-image under \theta
(1, 0) x - 1 + x+ y
(0, 1) x+ x - 1
(0, \eta ) x - 1 + y
| S\BbbF q(\gamma x)| = 2, now by using Theorem 2.1 and Lemma 2.1, we get
\BbbF qG
J(\BbbF qG)
\sim = \BbbF q \oplus \BbbF q2 .
Now suppose k is even, then we have
\BbbF qG
J(\BbbF qG)
\sim = \BbbF q \otimes \BbbF 2
\BbbF 2G
J(\BbbF 2G)
\sim = \BbbF q \otimes \BbbF 2 (\BbbF 2 \oplus \BbbF 4) \sim = \BbbF q \oplus \BbbF q2 . (2.1)
Theorem 2.5 is proved.
Corollary 2.2. The structure of \scrU
\biggl(
\BbbF qG
J(\BbbF qG)
\biggr)
is given by
\scrU
\biggl(
\BbbF qG
J(\BbbF qG)
\biggr)
\sim = Cq - 1 \oplus Cq2 - 1
and 1 + J(\BbbF qG) is a non-Abelian group of exponent 8.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1150 S. MAHESHWARI, R. K. SHARMA
Proof. We know that G has unique 2-Sylow subgroup of order 8. By using Proposition 2.3, we
have G \cap 1 + J(\BbbF qG) = OpG. Suppose that X = xy + x and Y = y2 + y, then X,Y \in J(\BbbF qG).
We see that XY \not = Y X, this proves that 1 + J(\BbbF qG) is a non-Abelian subgroup of \scrU (\BbbF qG). Now
by using Theorem 2.4, we have (1 + J(\BbbF qG))8 = 1. Since X4 \not = 0, it implies that (1 + J(\BbbF qG))
has exponent 8.
Theorem 2.6. Let p = 3, q = pk and G = SL(2,\BbbZ 3). Then
\BbbF qG
J(\BbbF qG)
\sim = \BbbF q \oplus \scrM (2,\BbbF q)\oplus \scrM (3,\BbbF q).
Proof. Suppose k is odd. We define the \BbbF q -algebra homomorphism
\theta \prime : \BbbF qG - \rightarrow \BbbF q \oplus \scrM (2,\BbbF q)\oplus \scrM (3,\BbbF q)
by an assignment
x \mapsto \rightarrow
\left( 1,
\Biggl[
- 1 - 1
1 0
\Biggr]
,
\left[
1 - 1 0
1 1 1
0 1 1
\right]
\right)
and
y \mapsto \rightarrow
\left( 1,
\Biggl[
0 - 1
1 0
\Biggr]
,
\left[
1 - 1 0
0 - 1 0
- 1 - 1 - 1
\right]
\right) .
Since \bigm| \bigm| S\BbbF q(\gamma y)
\bigm| \bigm| = \bigm| \bigm| S\BbbF q
\bigl(
\gamma y2
\bigr) \bigm| \bigm| = \bigm| \bigm| S\BbbF q(\gamma 1)
\bigm| \bigm| = 1.
Now by using Table 2, Theorem 2.1 and Lemma 2.1, we see that
\BbbF qG
J(\BbbF qG)
\sim = \BbbF q \oplus \scrM (2,\BbbF q)\oplus \scrM (3,\BbbF q).
Now suppose k is even, then apply the same argument as in equation (2.1). We get
\BbbF qG
J(\BbbF qG)
\sim = \BbbF q \oplus \scrM (2,\BbbF q)\oplus \scrM (3,\BbbF q).
Theorem 2.6 is proved.
Corollary 2.3. The structure of \scrU
\biggl(
\BbbF qG
J(\BbbF qG)
\biggr)
is given by
\scrU
\biggl(
\BbbF qG
J(\BbbF qG)
\biggr)
\sim = Cq - 1 \oplus GL(2,\BbbF q)\oplus GL(3,\BbbF q)
and 1 + J(\BbbF qG) is a non-Abelian group of exponent 3.
Proof. We can directly obtain the structure of \scrU
\biggl(
\BbbF qG
J(\BbbF qG)
\biggr)
, by Theorem 2.6. Observe that G is
p-solvable, so we have (1+J(\BbbF qG))3 = 1 is a group of exponent 3 as a consequence of Theorem 2.4.
Let X = - x - 1+y - y - 1 - yxy+y2x - 1+x - 1y - 1x - 1 and Y = - 1+y2+xy - yx - 1 - xy - 1+y - 1x - 1,
we can see that X,Y \in J(\BbbF qG). Further, XY \not = Y X, hence 1 + J(\BbbF qG) is a non-Abelian group.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
A NOTE ON UNITS IN \BbbF qSL(2,\BbbZ 3) 1151
Table 2. Ontoness of \theta \prime
Basis element Pre-image under \theta \prime
1, O,O - x - 1 - y + y2 - y - 1 - x - 1y + x - 1y2 + x - 1y - 1 - yx - 1 + y - 1x - 1 - xyx -
- xyx - 1 + xy - 1x - xy - 1x - 1 - x - 1yx - x - 1y - 1x
O,E2
1,1, O - x + x - 1 - y + y - 1 + xy - yx - xy2 + x - 1y + x - 1y2 + x - 1y - 1 - y - 1x -
- y - 1x - 1 - xy - 1x - x - 1yx+ x - 1y - 1x - 1 + x - 1y - 1x
O,E2
1,2, O 1+x - 1+x - 1y - 1x+xy2+x - 1y+xy - y - 1 - yx - x - 1yx - 1+y2+x - 1y - 1x - 1+
+ y - 1x - xyx - 1 + y - 1x - 1 + x - 1y - 1x - x+ xy - 1x - xy - 1
O,E2
2,1, O - yx + y - 1x2 - y2 - xy - 1x - 1 - x - x - 1y - 1 + xy - 1x + xyx - xy2 - x - 1 +
+ yx - 1 + x - 1yx - 1 + 1 + xy - y - 1x+ xyx - 1
O,E2
2,2, O - 1+x - y - 1 - yx - y2 - xy2 - x - 1 - y - 1 - x - 1y+y - 1x - y - 1x - 1 - xy - 1x -
- xyx - 1 - xy - 1x - 1 + x - 1y - 1x+ x - 1yx - 1 - x - 1y - 1x - 1
O,O,E3
1,1 - 1 - x+ x - 1 + y + y2 - y - 1 + xy - xy - 1 - x - 1y2 - x - 1y - 1 - yx - yx - 1 +
+ y - 1x+ y - 1x - 1 - xy - 1x - 1 - x - 1yx - 1 + x - 1y - 1x - 1
O,O,E3
1,2 - 1 - x - x - 1+y+y2 - y - 1 - xy - xy - 1 - x - 1y2 - x - 1y - 1+yx+yx - 1 - y - 1x -
- y - 1x - 1 - xyx - xyx - 1+xy - 1x - x - 1yx+x - 1yx - 1 - x - 1y - 1x - x - 1y - 1x - 1
O,O,E3
1,3 1 + x - 1 - y + y2 + y - 1 - xy2 - x - 1y + x - 1y - 1 + y - 1x - y - 1x - 1 - xyx -
- xyx - 1 - xy - 1x - xy - 1x - 1 - x - 1yx+ x - 1yx - 1 - x - 1y - 1x - 1
O,O,E3
2,1 1 - x + x - 1 + y - y2 + y - 1 + xy + xy2 + xy - 1 - x - 1y - x - 1y2 + yx - 1 -
- y - 1x+ xyx - xyx - 1 - xy - 1x - 1 - x - 1y - 1x+ x - 1yx+ x - 1y - 1x - 1
O,O,E3
2,2 - 1 + x - 1 - y - y2 + xy2 - xy - 1 + x - 1y + x - 1y - 1 - yx - yx - 1 - y - 1x +
+ y - 1x - 1 - xyx+ xyx - 1 + xy - 1x - x - 1yx - x - 1y - 1x - 1
O,O,E3
2,3 - y - 1 - xy - xy2 + x - 1y2 + x - 1y - 1 + x - 1y - yx - y - 1x+ y - 1x - 1 - xyx+
+ xy - 1x - 1 - x - 1yx+ x - 1y - 1x+ x - 1y - 1x - 1
O,O,E3
3,1 1 - y - y - 1 - xy - xy2 + xy - 1 - x - 1y2 + x - 1y - 1 - yx + yx - 1 - y - 1x +
+y - 1x - 1 - xyx+xy - 1x - xyx - 1+xy - 1x - 1 - x - 1yx+x - 1yx - 1 - x - 1y - 1x - 1
O,O,E3
3,2 1 - x+y - y - 1 - y2+xy - 1 - x - 1y+x - 1y2+yx+yx - 1+y - 1x - 1 - xyx - 1+
+ xyx - xy - 1x - 1 + x - 1yx - 1 - x - 1y - 1x+ x - 1y - 1x - 1
O,O,E3
3,3 1 + x - 1 - y - 1 - xy - xy2 + xy - 1 + x - 1y + yx - 1 + y - 1x+ y - 1x - 1 - xyx -
- xyx - 1 - xy - 1x - 1 - x - 1yx
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Received 15.12.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| spelling | umjimathkievua-article-5882025-03-31T08:47:35Z A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ A note on units in $F_q SL(2, Z_3)$ A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ Maheshwari , S. Sharma , R. K. MAHESHWARI, SWATI SHARMA, R. K. Maheshwari , S. Sharma , R. K. Group Algebra Unit Group Finite Field Group Algebra Unit Group Finite Field UDC 512.5 Let $R$ be a ring, and $ SL(2,R)$ be the special linear group of $2\times2$ matrices with determinant $1$ over $R$. We obtain the Wedderburn decomposition of$\dfrac{\mathbb{F}_q SL(2,\mathbb{Z}_3)}{J(\mathbb{F}_q SL(2,\mathbb{Z}_3))}$ and show that $ 1+J(\mathbb{F}_q SL(2,\mathbb{Z}_3))$ is a non-Abelian group, where $\mathbb{F}_q$ is a finite field with $q = p^k$ elements of characteristic $2$ and $3.$   УДК 512.5 Повідомлення про одиниці у $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ Нехай $R$ - кільце, а $ SL(2,R)$ -спеціальна лінійна група$(2\times 2)$-матриць над $R$ з детермінантом $1.$ Отримано декомпозицію Веддербурна для$\dfrac{\mathbb{F}_q SL(2,\mathbb{Z}_3)}{J(\mathbb{F}_q SL(2,\mathbb{Z}_3))}$ і показано, що$1+J(\mathbb{F}_q SL(2,\mathbb{Z}_3))$ є неабелевою групою, де $\mathbb{F}_q$ -скінченне поле з $q = p^k$ елементами та характеристикою $2$ або $3.$ Institute of Mathematics, NAS of Ukraine 2021-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/588 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 8 (2021); 1147 - 1152 Український математичний журнал; Том 73 № 8 (2021); 1147 - 1152 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/588/9101 |
| spellingShingle | Maheshwari , S. Sharma , R. K. MAHESHWARI, SWATI SHARMA, R. K. Maheshwari , S. Sharma , R. K. A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title | A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title_alt | A note on units in $F_q SL(2, Z_3)$ A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title_full | A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title_fullStr | A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title_full_unstemmed | A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title_short | A note on units in $\mathbb{F}_q SL(2, \mathbb{Z}_3)$ |
| title_sort | note on units in $\mathbb{f}_q sl(2, \mathbb{z}_3)$ |
| topic_facet | Group Algebra Unit Group Finite Field Group Algebra Unit Group Finite Field |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/588 |
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