arXiv subjects
Publications and source records attributed to Hemant Kalra.
A long-standing conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we settle the conjecture for a finite $p$-group ($p >2$) of nilpotency class $n$ with certain conditions.
In this note, we classify all finite groups having exactly 6, 7 or 8 cyclic subgroups. This gives a partial answer to the open problem posed by Tarnauceanu (Amer. Math. Monthly, 122 (2015), 275-276). As a consequence of our results, we also obtain an important result concerning with the converse of Lagrange's theorem.
It is proved in [J. Group Theory, {\bf 10} (2007), 859-866] that if $G$ is a finite $p$-group such that $(G,Z(G))$ is a Camina pair, then $|G|$ divides $|\Aut(G)|$. We give a very short and elementary proof of this result.
Let $G$ be a finite $p$-group and let Aut$(G)$ denote the full automorphism group of $G$. In the recent past, there has been interest in finding necessary and sufficient conditions on $G$ such that certain subgroups of Aut$(G)$ are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.
Let $G$ be a finite group. Let $k(G)$ denote the number of conjugacy classes of $G$ and let $m(G)$ denote the least positive integer $n$ such that the union of any $n$ distinct non-trivial conjugacy classes of $G$ together with the identity of $G$ is a subgroup of $G$. We prove that $m(G)=k(G)-1$ for all $m(G)\ge 2$.
Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. Let $\mathrm{Aut}_c(G)$ and $\mathrm{Aut}_z(G)$ respectively denote the group of all class preserving and central automorphisms of $G$. We give a necessary condition for $G$ such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and give necessary and sufficient conditions for $G$ with elementary abelian or cyclic center such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G).$ We also characterize all finite $p$-groups $G$ of order $\leq p^7$ such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and complete the classification of all finite $p$-groups of order $\le p^5$ for which there exist non-inner class preserving automorphisms.
We study finite p-groups G of coclass upto 4 for which the group Aut_z(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].
An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results.
We study class preserving automorphisms of groups of order thirty two and prove that only two groups have non-inner class preserving automorphisms.