Schur multiplier exponents of finite p-groups: bounds and critical-class phenomena

The Schur exponent conjecture is a long-standing conjecture in group theory, which asserts that exp(M(G))(G) for every finite group G, where M(G) denotes the Schur multiplier of G. In this article, we prove that the Schur exponent conjecture holds for finite p-groups G of nilpotency class at most $2p-2$ when p is an odd prime. Our approach is to establish upper bounds on the exponent of γₛ₊₁(G) for s. Using this approach, we further establish exponent bounds for Schur multipliers, nonabelian exterior squares, and tensor squares of G of arbitrary nilpotency class, improving some existing bounds. Moreover, we show that Moravec's Bogomolov multiplier exponent conjecture holds for finite p-groups of nilpotency class at most $2p-1$. Havas conjectured that, for every prime $p>3$, there exists a finite p-group of exponent p and nilpotency class $2p-1$ for which the Schur exponent conjecture fails. We show that the Schur exponent conjecture still holds for finite p-groups of nilpotency class at most $2p-1$ generated by at most three elements. For four-generator groups of exponent p and class at most $2p-1$, we prove that M(G) Cp²\,b× Cₚ\,a for some a≥0 and b∈\0,1\.