Bounds in the restricted Burnside problem

Abstract

We survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of Pi-algebras.1991 Mathematics subject classification {Amer. Math. Soc): primary 20F05, 20D15.

Finite Groups of Bounded Exponent

“…This improves upon the bound of T (m, n n n ) found by Vaughan-Lee and Zelmanov [7]. For an overview of issues about bounds in the restricted Burnside problem, see [8].…”

Groups of small period growth

“…Since Q2 excludes finite groups, this function yields a lower bound for the period growth function of any m-generated residually finite infinite p-group. The best known lower bound for zel m (n) is due to Groves and Vaughan-Lee [6], who prove that ) ≤ 2 k , with k appearances of the number 2 in the tower on the left side, which is due to Newman, whose argument is given in [10].…”

Invariants for metabelian groups of prime power exponent, colorings, and stairs

“…The order of for arbitrary d is still unknown. Upper and lower bounds for have been given by Vaughan-Lee and Zel’manov [50–52] and by Groves and Vaughan-Lee [16].…”

“…For each d and each prime power n, the p-quotient algorithm can be used to obtain a consistent polycyclic/power-commutator presentation of R(d, n) and then to compute its order (see [45,Section 11.7], [41,49]); however, a direct implementation of this algorithm takes in general too much time. Havas et al [25] proved that |R(2, 5)| = 5 34 , O'Brien and Vaughan-Lee [41] proved that |R(2, 7)| = 7 20416 , and Newman and O'Brien [39] proved that |B (5,4) [50][51][52] and by Groves and Vaughan-Lee [16].…”