Type: Seminar

Distribution: World

Expiry: 29 Aug 2013

Auth: brendan@60-242-80-51.static.tpgi.com.au (bcreutz) in SMS-WASM

Date: 29/08/13 Location: Carslaw 535 Time: 15:00 Name: Manoj Yadav Affiliation: Harish-Chandra Research Institute Title: Central factor and commutator subgroup of nilpotent groups Abstract: 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 Schur’s theorem. The converse ofthis result is not true in general as shown by infinite extra-special $p$-groups for odd primes. But when $G$ is finitely generated by $d$ elements (say) and $\gamma_2(G)$ is finite, it was proved by B. H. Neumann that $|G/Z(G)| \le |\gamma_2(G)|^d$. In this talk I start with this basic result, discuss a slight generalization and classify all non-abelian finite nilpotent groups $G$ minimally generated by $d := d(G)$ elements such that central quotient is maximal, i.e., $|G/Z(G)| = |\gamma_2(G)|^d$. First I reduce the problem to finite $p$-groups and then notice that such $p$-groups admit maximum number of (conjugacy) class-preserving automorphisms. Using this and some basic Lie theoretic techniques, I show that $d(G)$ is even. Moreover, when the nilpotency class of $G$ is at least $3$, then $d(G) = 2$. The situation is then divided into two cases, depending on whether $p$ is even (Magma is extremely useful in this case) or odd, and a complete classification is presented. All $p$-groups $G$ considered above satisfy the following condition: $|x^G| = |\gamma_2(G)|$ for all $x \in G - \Phi(G)$, where $x^G$ denotes the conjugacy class of $x$ in $G$ and $\Phi(G)$ denotes the Frattini subgroup of $G$. Let me call such a group Camina-type. One of my motives to visit the Computational Algebra group here is to find a solution to the following problem: Does there exist a finite Camina-type $p$-group $G$, $p$ odd, of nilpotency class at least $3$ such that $\gamma_2(G) < \Phi(G)$ and $\gamma_2(G)$ is non-cyclic?