Speaker: Dr Renu Joshi, IISER Bhopal
Time, Day and Date: 2.30 pm, Thursday, October 3, 2024.
Google Meet Link: https://meet.google.com/fqv-owns-ydv
Title: The Schur Multiplier and Non-inner Automorphism Conjecture
Abstract: The Schur multiplier M(G) of a finite group G is defined as H^2(G, ℂ×), where ℂ× is the trivial G-module. Several authors have determined upper bounds on the order, rank, and exponent of M(G). If G is a p-group of a maximal class of order p^n and 4 ≤ n ≤ p+1, we demonstrate that the Schur multiplier M(G) is an elementary abelian p-group. Additionally, the case n = p+1 answers a question raised by Primož Moravec.
In 1973, Berkovich asked if every finite non-abelian p-group has a non-inner automorphism of order p, for any prime p. This problem is solved in the positive direction for several classes of p-groups. In 2017, it was proven for p-groups of co-class 3 for p≠3. Consequently, the following question arises: Does the non-inner automorphism conjecture hold for finite 3-groups of co-class 3? Our current work focuses on 3-groups of co-class 3. We will discuss our progress and findings in this direction.