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For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\Aut \, (G, \cdot)$, which, in general, may not be a homomorphism. A skew left brace will be called $\lambda$-anti-homomorphic ($\lambda$-homomorphic) if $\lambda : (G, \cdot) \to \Aut \, (G, \cdot)$ is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitute a $\lambda$-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite sets., Comment: 25 pages

We present a characterisation of groups $G$ of order $p^7$, $p$ prime, in which not all elements of the commutator subgroup $\gamma_2(G)$ of $G$ are commutators in $G$. On the way we obtain several structural results on groups of order $p^7$., Comment: 17 pages

The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial braces is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed., Comment: 24 pages, Comments are welcome

COVID-19--a viral infectious disease--has quickly emerged as a global pandemic infecting millions of people with a significant number of deaths across the globe. The symptoms of this disease vary widely. Depending on the symptoms an infected person is broadly classified into two categories namely, asymptomatic and symptomatic. Asymptomatic individuals display mild or no symptoms but continue to transmit the infection to otherwise healthy individuals. This particular aspect of asymptomatic infection poses a major obstacle in managing and controlling the transmission of the infectious disease. In this paper, we attempt to mathematically model the spread of COVID-19 in India under various intervention strategies. We consider SEIR type epidemiological models, incorporated with India specific social contact matrix representing contact structures among different age groups of the population. Impact of various factors such as presence of asymptotic individuals, lockdown strategies, social distancing practices, quarantine, and hospitalization on the disease transmission is extensively studied. Numerical simulation of our model is matched with the real COVID-19 data of India till May 15, 2020 for the purpose of estimating the model parameters. Our model with zone-wise lockdown is seen to give a decent prediction for July 20, 2020., Comment: 34 Pages, 14 Figures, 2 Tables

We present a classification of finite $p$-groups $G$ with $\gamma_2(G)$, the commutator subgroup of $G$, of order $p^4$ and exponent $p$ such that not all elements of $\gamma_2(G)$ are commutators., Comment: 28 pages. Included the case $p=2$ and corrected some ambiguities

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \mathrm{Aut} \;(G, \cdot),~~a \mapsto \lambda_a$, where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\mathrm{Aut}\; (G, \cdot)$, which, in general, may not be a homomorphism. We study skew left braces $(G, \cdot, \circ)$ for which $\lambda : (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ is a homomorphism. Such skew left braces will be called $\lambda$-homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism $\lambda : (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ gives rise to a skew left brace, which, indeed, is $\lambda$-homomorphic. As an application, we construct skew left braces when $(G, \cdot)$ is either a free group or a free abelian group. We prove that any $\lambda$-homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on $\lambda$-homomorphic skew left brace for which the image of $\lambda$ is cyclic. A complete characterization of such skew left braces on the free abelian group of rank two is obtained., Comment: 23 pages, Comments are welcome

We improve Algorithm 5.1 of [Math. Comp. {\bf 86} (2017), 2519-2534] for computing all non-isomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we state some conjectures for further research., Comment: 11 pages, 8 tables, Computational results

Let $q$ be a $p$-power where $p$ is a fixed prime. In this paper, we look at the $p$-power maps on unitriangular group $U(n,q)$ and triangular group $T(n,q)$. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian $p$-groups of order $p^5$. As an application we determine the capability of groups of order $p^5$., Comment: 41 pages, 5 tables

Let $G$ be a central product of two groups $H$ and $K$. We study second cohomology group of $G$, having coefficients in a divisible abelian group $D$ with trivial $G$-action, in terms of the second cohomology groups of certain quotients of $H$ and $K$. In particular, for $D = \mathbb{C}^{*}$, some of our results provide a refinement of results from [Some groups with non-trivial multiplicators, Math. Z. {\bf 120 } (1971), 307-308] and [On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra {\bf 3} (1973), 73-82]., Comment: 10 pages, to appear in Journal of Pure and Applied Algebra

It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall)., Comment: 24 pages, Commets are welcome, some anomalies in the first version have been corrected

In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.

We classify finite $p$-groups, upto isoclinism, which have only two conjugacy class sizes $1$ and $p^3$. It turns out that the nilpotency class of such groups is $2$., Comment: 12 pages

The non-abelian groups with abelian group of automorphisms are widely studied. Following Earnley, such groups are called Miller groups, since the first example of such a group was given by Miller in 1913. Many other examples of Miller $p$-groups have been constructed by several authors. Recently, A. Caranti [{\it Israel J. Mathematics {\bf 205} (2015), 235-246}] provided module theoretic methods for constructing non-special Miller $p$-groups from special Miller $p$-groups. By constructing examples, we show that these methods do not always work. We also provide a sufficient condition on special Miller $p$-group for which the methods of Caranti work.

In this paper we study the existence of at least one non-inner automorphism of order p of a non-abelian finite p-group of coclass 3, where p is a prime integer such that p is different from 3.

We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115} (1992), 955-959] and Dolfi et al. [Israel J. Math. {\bf 198} (2013), 283-331].

Let $G$ be a group and $Out_c(G)$ be the group of its class-preserving outer automorphisms. We compute $|Out_c(G)|$ for all the group $G$ of order $p^6$, where $p$ is an odd prime. As an application, we observe that if $G$ is a $\CYRSH$-rigid group of order $p^6$, then it's Bogomolov multiplier $B_0(G)$ is zero., Comment: 12 pages

Let $G$ be a finite group minimally generated by $d(G)$ elements and $\Aut_c(G)$ denote the group of all (conjugacy) class-preserving automorphisms of $G$. Continuing our work [Class preserving automorphisms of finite $p$-groups, J. London Math. Soc. \textbf{75(3)} (2007), 755-772], we study finite $p$-groups $G$ such that $|\Aut_c(G)| = |\gamma_2(G)|^{d(G)}$, where $\gamma_2(G)$ denotes the commutator subgroup of $G$. If $G$ is such a $p$-group of class $2$, then we show that $d(G)$ is even, $2d(\gamma_2(G)) \le d(G)$ and $G/\Z(G)$ is homocyclic. When the nilpotency class of $G$ is larger than $2$, we obtain the following (surprising) results: (i) $d(G) = 2$. (ii) If $|\gamma_2(G)/\gamma_3(G)| > 2$, then $|\Aut_c(G)| = |\gamma_2(G)|^{d(G)}$ if and only if $G$ is a $2$-generator group with cyclic commutator subgroup, where $\gamma_3(G)$ denotes the third term in the lower central series of $G$. (iii) If $|\gamma_2(G)/\gamma_3(G)| = 2$, then $|\Aut_c(G)| = |\gamma_2(G)|^{d(G)}$ if and only if $G$ is a $2$-generator $2$-group of nilpotency class $3$ with elementary abelian commutator subgroup of order at most $8$. As an application, we classify finite nilpotent groups $G$ such that the central quotient $G/\Z(G)$ of $G$ by it's center $\Z(G)$ is of the largest possible order. For proving these results, we introduce a generalization of Camina groups and obtain some interesting results. We use Lie theoretic techniques and computer algebra system `Magma' as tools., Comment: 30 pages

We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $\gamma_2(G) < \mathrm{Z}(G) < \Phi(G)$, where $\gamma_2(G)$, $\mathrm{Z}(G)$ and $\Phi(G)$ denote the commutator subgroup, the center and the Frattini subgroup of $G$ respectively. For a finite $p$-group $G$ with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) $\mathrm{Z}(G) = \Phi(G)$ is elementary abelian; (ii) $\gamma_2(G) = \Phi(G)$ is elementary abelian, where $p$ is an odd prime. We construct examples to show the existence of groups $G$ with elementary abelian automorphism group for which exactly one of the above two conditions holds true., Comment: 13 pages, 2 tables. Accepted for publication in International Journal of Algebra and Computation. arXiv admin note: text overlap with arXiv:1005.2066

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting classes of groups having only two different conjugacy class sizes. We re-prove the fact that if $G$ is a finite group such that the set of its conjugacy class sizes is $\{1, p\}$, where $p$ is a prime integer, then $G$ is isoclinic (in the sense of P. Hall) to an extraspecial $p$-group., Comment: 14 pages

Let $G$ be an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$. Then $\gamma_2(G)$, the commutator subgroup of $G$, is finite. This result is known as Shur's theorem (the Schur's theorem). In this short note we provide a quick survey on the converse of Schur's theorem, generalize known results in this direction and prove the following result (which is perhaps the most suitable statement for converse of the Schur's theorem): If $G$ is an arbitrary group with finite $\gamma_2(G)$, then $G/\Z(G)$ is finite if $\Z_2(G)/\Z(\Z_2(G))$ is finitely generated, where $\Z_2(G)$ denotes the second center of a group $G$. If $G/\Z(G)$ is finite, then $\gamma_2(G)$ is also finite and $|G/\Z(G)| \le |\gamma_2(G)|^d$, where $d$ denotes the number of elements in any minimal generating ser for $G/\Z(G)$. We classify all nilpotent groups $G$ of class 2 upto isoclinism (in the sense of P. Hall) such that $|G/\Z(G)| = |\gamma_2(G)|^d$, and ask some questions in the sequel., Comment: Contents published in arXiv:1011.2083

We obtain certain results on a finite $p$-group whose central automorphisms are all class preserving. In particular, we prove that if $G$ is a finite $p$-group whose central automorphisms are all class preserving, then $d(G)$ is even, where $d(G)$ denotes the number of elements in any minimal generating set for $G$. As an application of these results, we obtain some results regarding finite $p$-groups whose automorphisms are all class preserving. In particular, we prove that if $G$ is a finite $p$-groups whose automorphisms are all class preserving, then order of $G$ is at least $p^8$ and the order of the automorphism group of $G$ is at least $p^12$., Comment: 12 pages, Accepted for publication in Comm. Algebra, A minor modification is done in the proof of Proposition 3.4 to make it work for all primes (including 2)