arXiv subjects
Publications and source records attributed to Meena Sahai.
Let $RG$ be the group ring of an arbitrary group $G$ over an associative non-commutative ring $R$ with identity. In this paper, we have obtained the necessary and sufficient conditions under which $RG$ is Jordan nilpotent of index $4$.
Let $C_n$, $Q_n$ and $D_n$ be the cyclic group, the quaternion group and the dihedral group of order $n$, respectively. The structures of the unit groups of the finite group algebras $FQ_{12}$ and $F(C_2 \times Q_{12})$ over a finite field $F$ have been studied in J. Gildea, F. Monaghan (2011), F. Monaghan (2012), G. Tang, Y. Gao (2011) and G. Tang, Y. Wei, Y. Li (2014) whereas the structures of the unit groups of the finite group algebras $FD_{12}$ and $F(C_2 \times D_{12})$ have been studied in J. Gildea, F. Monaghan (2011), N. Makhijani, R. K. Sharma, J. B. Srivastava (2016), F. Monaghan (2012), M. Sahai, S. F. Ansari and G. Tang, Y. Gao (2011). In this paper, we continue this study 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.
Let $D_{n}$ be the dihedral group of order $n$. The structure of the unit group $U(F(C_3 \times D_{10}))$ of the group algebra $F(C_3 \times D_{10})$ over a finite field $F$ of characteristic $3$ is given in \cite{sh13}. In this article, the structure of $U(F(C_3 \times D_{10}))$ is obtained over any finite field $F$ of characteristic $p \neq 3$.
In this paper, we classify the modular group algebra $KG$ of a group $G$ over a field $K$ of characteristic $p>0$ having upper Lie nilpotency index $t^{L}(KG)= \vert G^{\prime}\vert - k(p-1) + 1$ for $k=14$ and $15$. Group algebras of upper Lie nilpotency index $\vert G^{\prime}\vert - k(p-1) + 1$ for $k\leq 13$, have already been characterized completely.
In this article, we show that if $KG$ is Lie nilpotent group algebra of a group $G$ over a field $K$ of characteristic $p>0$, then $t_{L}(KG)=k$ if and only if $t^{L}(KG)=k$, for $k\in\{5p-3, 6p-4\}$, where $t_{L}(KG)$ and $t^{L}(KG)$ are the lower and the upper Lie nilpotency indices of $KG$, respectively.
Let $F$ be a finite field of characteristic $p$. The structures of the unit groups of group algebras over $F$ of the three groups $D_{24}$, $S_4$ and $SL(2, \mathbb{Z}_3)$ of order $24$ are completely described in \cite{K4, SM, SM1, FM, sh1}. In this paper, we give the unit groups of the group algebras over $F$ of the remaining groups of order $24$, namely, $C_{24}$, $C_{12} \times C_2$, $C_2^3 \times C_3$, $C_3 \rtimes C_8$, $C_3 \rtimes Q_8$, $D_6 \times C_4$, $C_6 \rtimes C_4$, $C_3 \rtimes D_8$, $C_3 \times D_8$, $C_3 \times Q_8$, $A_4 \times C_2$ and $D_{12} \times C_2$.