Let G be a finite group. Let $D =\{(g, g)| g\in G\}$, the main diagonal subgroup of $G\times G$. In this paper, we consider the suitable generalized normalities or index of D in $G\times G$, some interesting results are obtained.

In this paper, the concepts of $\mathscr {P}$-condensing operator and $\mathscr {C}\mathscr {P}$-condensing operator on a semigroup are introduced. Then we describe the $\mathscr {P}$-condensing operators and the $\mathscr {C}\mathscr {P}$-condensing operators on semilattices, join-complete lattices, and inverse semigroups. As an application of our results, we also show that the unary operations${}^\dag $: $Hg\mapsto H\wedge g^{-1}Hg$ and ${}^\ddag $: $Hg\mapsto H_G$ on the coset semigroup $\mathbb {K}(G)$ of a group G are $\mathscr {P}$-condensing operators. Consequently, a necessary and sufficient condition for these operators to be $\mathscr {C}\mathscr {P}$-condensing operators is given. Some further results on condense semigroups considered by Chen et al. (Acta Math. Hung. 119: 281–305, 2008) are discussed and obtained in this paper.

Suppose that A is a subgroup of a group G. A is called to be m-embedded in G if G has a subnormal subgroup T and a $\{1\le G\}$-embedded subgroup C such that $G=AT$ and $A\cap T\le C\le A$. In this paper, we shall investigate the structure of finite groups by using m-embedded subgroups and obtain some new characterization about p-supersolvability and generalized hypercentre of finite groups. Some results in Guo and Shum (Arch Math 80:561–569, 2003) , Ramadan et al. (Arch Math 85:203–210, 2005), Tang and Miao (Turk J Math 39:501–506, 2015), and Xu and Zhang (Can Math Bull 57(4):884–889, 2014) are generalized.

The Laplace distribution can be compared against the normal distribution. The Laplace distribution has an unusual, symmetric shape with a sharp peak and tails that are longer than the tails of a normal distribution. It has recently become quite popular in modeling financial variables (Brownian Laplace motion) like stock returns because of the greater tails. The Laplace distribution is very extensively reviewed in the monograph (Kotz et al. in the laplace distribution and generalizations—a revisit with applications to communications, economics, engineering, and finance. Birkhauser, Boston, 2001). In this article, we propose a density-based empirical likelihood ratio (DBELR) goodness-of-fit test statistic for the Laplace distribution. The test statistic is constructed based on the approach proposed by Vexler and Gurevich (Comput Stat Data Anal 54:531–545, 2010). In order to compute the test statistic, parameters of the Laplace distribution are estimated by the maximum likelihood method. Critical values and power values of the proposed test are obtained by Monte Carlo simulations. Also, power comparisons of the proposed test with some known competing tests are carried out. Finally, two illustrative examples are presented and analyzed.

In this article, an additive Perks–Weibull model capable of modeling lifetime data with bathtub-shaped hazard rate function is proposed. The model is derived by the sum of the hazard rates of Perks and Weibull distributions. Some statistical properties including shapes of density and hazard rate functions, moments, and order statistics are explored. The method of maximum likelihood estimation is used for estimating the model parameters. The goodness-of-fit of the model for three real datasets having bathtub-shaped hazard rate functions has been illustrated. Finally, an application for competing risk data is also given to show the flexibility of the proposed model.

In this paper, we study such polyadic analog of an identity of a group as m-neutral sequence. In particular, we prove that all Post’s equivalence classes of the free covering group of any n-ary group [where $n = k(m - 1) + 1$ and $k\ge 1$] defined by m-neutral sequences form the $(k + 1)$-ary group, which is isomorphic to the n-ary subgroup of all identities of the n-ary group in the case when $m = 2$.