arXiv subjects
Publications and source records attributed to Palash Sharma.
Let $G$ be a finite group. Let $C(G)$ be the set of cyclic subgroups of $G$, $c(G)=|C(G)|$ and $i(G)=\left|\{x\in G : x^{2}=e\}\right|$. In this article, we classify finite groups with $i(G)=c(G)-r$ for $r\in \{0,1,2\}$ and prove that the range of the function given by $\beta(G)=\frac{i(G)}{c(G)}$ is dense in $[0,1]$. We also answer an open question posed by Gao and Shen in `Finite groups with many cyclic subgroups'.
Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American Mathematical Monthly 122.3 (2015): 275-276], an open problem was asked to classify the groups $G$ with $|C(G)|=|G|-r$, where $2 \leq r \leq |G|-1$. In this article, first we show that, for an odd prime $p$, there are infinitely many groups $G$ with $|C(G)|= \frac{|G|}{2}$, $|C(G)|=\frac{|G|}{p^{q-1}}$ (for prime $q\neq p)$, or $|C(G)|=\frac{|G|}{2}+2^{k}, k\geq 0$. Then, we partially answer the open question by classifying finite groups $G$ having $\frac{|G|}{2}-1\leq |C(G)| \leq \frac{|G|}{2}+1$ for some fix values of $\mathcal{O}_{2}(G)$. Finally, we provide a complete list of finite groups $G$ having $|C(G)|=\frac{|G|+(2r+1)}{2}$ for $r\geq-1$.
This work provides a survey of the general class of distributions generated from the mixture of the beta random variables. We provide an extensive review of the literature, concerning generating new distributions via the inverse CDF transformation. In particular, we accounted for beta generated and Kumaraswamy generated families of distributions. We provide a brief summary of each of their families of distributions. We also propose a new asymmetric mixture distribution, which is an alternative to beta generated distributions. We provide basic properties of this new class of distributions generated from the Laplace model. We also address the issue of parameter estimation of this new skew generalized Laplace model.