arXiv subjects
Publications and source records attributed to Manoj K. Yadav.
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.
In 1904, Issai Schur proved the following result. If $G$ is an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$, then the commutator subgroup of $G$ is finite. A partial converse of this result was proved by B. H. Neumann in 1951. He proved that if $G$ is a finitely generated group with finite commutator subgroup, then $G/\Z(G)$ is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups $G$ such that $|G/\Z(G)| = |γ_2(G)|^d$, where $d$ denotes the number of elements in a minimal generating set for $G/\Z(G)$. Some problems and questions are posed in the sequel.
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.
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 $|\Aut_c(G)| = |γ_2(G)|^{d(G)}$, where $γ_2(G)$ denotes the commutator subgroup of $G$. If $G$ is such a $p$-group of class $2$, then we show that $d(G)$ is even, $2d(γ_2(G)) \le d(G)$ 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 $|γ_2(G)/γ_3(G)| > 2$, then $|\Aut_c(G)| = |γ_2(G)|^{d(G)}$ if and only if $G$ is a $2$-generator group with cyclic commutator subgroup, where $γ_3(G)$ denotes the third term in the lower central series of $G$. (iii) If $|γ_2(G)/γ_3(G)| = 2$, then $|\Aut_c(G)| = |γ_2(G)|^{d(G)}$ 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.
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 $p^12$.
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 $γ_2(G) < \mathrm{Z}(G) < Φ(G)$, where $γ_2(G)$, $\mathrm{Z}(G)$ and $Φ(G)$ 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) $\mathrm{Z}(G) = Φ(G)$ is elementary abelian; (ii) $γ_2(G) = Φ(G)$ 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.
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.
We classify all finite $p$-groups $G$ for which |$Aut_{c}(G)$| attains its maximum value, where $Aut_{c}(G)$ denotes the group of all class preserving automorphisms of $G$ .
In this short survey article, we try to list maximum number of known results on class preserving automorphisms of finite $p$-groups. We conclude the survey with some interesting (at least for the author) open problems on this topic.
Let $\textrm{UT}_n (K)$ be a unitriangular group over a field $K$ and $Γ_{n,k} := \textrm{UT}_n (K)/ γ_k(\textrm{UT}_n (K))$, where $γ_k (\mathrm{UT}_n(K))$ denotes the $k$-th term of the lower central series of $\mathrm{UT}_n (K)$, $2 \le k \le n$. We prove that the group of all class preserving automorphisms of $Γ_{n,k}$ is equal to $\Inn(Γ_{n,k})$ if and only if $K$ is a prime field. Let $G_n^{(m)} := \mathrm{UT}_n (\mathbb{F}_{p^m}) / γ_3 (\mathrm{UT}_n(\mathbb{F}_{p^m}))$. We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of $G_n^{(m)}$.
An automorphism $α$ of a group $G$ is said to be central if $α$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter examples to a conjecture of A. Mahalanobis [Israel J. Math., {\bf 165} (2008), 161 - 187]. We also construct a family of finite $p$-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in group theory, Aracne Editrice, Rome 2002, 111-127].
We give a sufficient condition on a finite $p$-group $G$ of nilpotency class 2 so that $\Aut_c(G) = \Inn(G)$, where $\Aut_c(G)$ and $\Inn(G)$ denote the group of all class preserving automorphisms and inner automorphisms of $G$ respectively. Next we prove that if $G$ and $H$ are two isoclinic finite groups (in the sense of P. Hall), then $\Aut_c(G) \cong \Aut_c(H)$. Finally we study class preserving automorphisms of groups of order $p^5$ and prove that $\Aut_c(G) = \Inn(G)$ for all the groups of order $p^5$ except two isoclinism families.
Let $G$ be a finite group and $M(G)$ be the subgroup of $G$ generated by all non-central elements of $G$ that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of $M(G)$ and $F(M(G))$, where $F(M(G))$ denotes the Fitting subgroup of $M(G)$. We prove some conditional results regarding the nilpotency class of $M(G)$.
We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $γ_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.
Let $1 \to N \to G \to H \to 1$ be an abelian extension. The purpose of this paper is to study the problem of extending automorphisms of $N$ and lifting automorphisms of $H$ to certain automorphisms of $G$.
Let $G$ be a finite $p$-group of nilpotency class 2. We find necessary and sufficient conditions on $G$ such that each central automorphism of $G$ fixes the center of $G$ element-wise.
Let $G$ be a finite $p$-group such that $x\Z(G) \subseteq x^G$ for all $x \in G- \Z(G)$, where $x^G$ denotes the conjugacy class of $x$ in $G$. Then $|G|$ divides $|\Aut(G)|$, where $\Aut(G)$ is the group of all automorphisms of $G$.