The unit group of finite group algebra of a generalized dihedral group

Makhijani, Neha; Sharma, Rajendra K.; Srivastava, J. B.

doi:10.1142/s179355711450034x

Cited by 7 publications

(3 citation statements)

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“…The set of all invertible elements of F q G form a group called the unit group of F q G denoted by U (F q G). There are many known results about the group unit of F q G in [2,3,4]. The main topic of this study is to find the nullity degree (probability) of F q G, which is the probability that the multiplication of two randomly chosen elements of F q G is zero.…”

Section: Introductionmentioning

confidence: 99%

The probability that the multiplication of two ring elements is zero

Amiri

2018

J. Algebra Appl.

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Let [Formula: see text] be a finite commutative ring. We denote by [Formula: see text] the probability that the multiplication of two randomly chosen elements of [Formula: see text] is zero. In this paper, we show that either [Formula: see text] or [Formula: see text] for any ring [Formula: see text] and determine all rings [Formula: see text] with [Formula: see text]. Also, we obtain the structures of rings [Formula: see text] having maximum or minimum value of [Formula: see text] among all rings with identity of the same size. We characterize all rings [Formula: see text] with identity such that [Formula: see text]. Finally, we compute [Formula: see text] where [Formula: see text] is a PIR local ring.

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Section: Introductionmentioning

confidence: 99%

The probability that the multiplication of two ring elements is zero

Amiri

2018

J. Algebra Appl.

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“…In addition to this, the recent counterexample to the renowned Kaplansky's unit conjecture further emphasizes the need of research in this area (see [11]). It has been extensively investigated how the unit group of the group algebra F q G is structured (see [1,3,4,[16][17][18]20,21,23,25,27,28]). Furthermore, there have been significant developments in the exploration of the unit group of modular group algebras, in addition to integral and semisimple group algebras (see [5][6][7][8] and the references therein for a comprehensive and recent literature in this direction).…”

Section: Introductionmentioning

confidence: 99%

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Sivaranjani,

Nandakumar,

Mittal

et al. 2024

Armen. J. Math.

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In this paper, we consider the general linear group $GL(2, 7)$ of $2 \times 2$ invertible matrices over the finite field of order $7$ and compute the unit group of the semisimple group algebra $\mathbb{F}_qGL(2,7)$, where $\mathbb{F}_q$ is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of $\mathbb{F}_qGL(2,7)$, which is determined by first computing the Wedderburn decomposition of the group algebra $\mathbb{F}_q(PSL(3, 2)\rtimes C_2)$. Here $PSL(3,2)$ is the projective special linear group of degree 3 over a finite field of 2 elements.

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“…[15, Theorem 2.1] Let F be a field of characteristic p having q = p k elements. If (n, p) = 1, where n ∈ N, then…”

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confidence: 99%

Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$

Sahai

Ansari

2023

International Electronic Journal of Algebra

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Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral group of order $n$, respectively. Recently, the structures of the unit groups of the finite group algebras of $2$-groups that contain a normal cyclic subgroup of index $2$ have been studied. The dihedral groups $D_{2n}, n\geq 3$ and the generalized quaternion groups $Q_{4n}, n\geq 2$ also contain a normal cyclic subgroup of index $2$. The structures of the unit groups of the finite group algebras $FQ_{12}$, $FD_{12}$, $F(C_2 \times Q_{12})$ and $F(C_2 \times D_{12})$ over a finite field $F$ are well known. In this article, we continue this investigation and establish the structures of the unit groups of the group algebras $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$ over a finite field $F$ of characteristic $p$ containing $p^k$ elements.

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The unit group of finite group algebra of a generalized dihedral group

Cited by 7 publications

References 10 publications

The probability that the multiplication of two ring elements is zero

The probability that the multiplication of two ring elements is zero

Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$

Units in $F(C_n \times Q_{12})$ and $F(C_n \times D_{12})$

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