Lower estimates of the order of maximal periodic groups of prime period
The authors obtain lower bounds for the order and the nilpotency class of \(\bar B(2,p)\)- the largest 2-generator finite group of prime exponent p (such a group exists via Kostrikin's positive solution of the restricted Burnside problem for groups of prime exponent. The lower bounds obtained are exponential with respect to p; namely, for sufficiently large p the nilpotency class of \(\bar B(2,p)\) is greater than \(2^{p/15,4}\) and \(\log_ p| \bar B(2,p)| >2^{2^{p/15,5}}\). The proof deals with the associated Lie ring \(L(\bar B(2,p))\) of \(\bar B(2,p)\). The authors represent \(L(\bar B(2,p))\) as L/I, where L is a free Lie ring and make use of \textit{M. R. Vaughan-Lee}'s recent description of the ideal I [from Bull. Lond. Math. Soc. 17, 113-133 (1985; Zbl 0544.20032)] in some ``associative'' terms. (Note that both the nilpotency class and the order of \(L(\bar B(2,p))\) and \(\bar B(2,p)\) coincide). The well-known Magnus- Sanov theorem gives only \(I\supset E_{p-1}+pL\), where \(E_{p-1}\) is the \((p-1)\)-Engel ideal of L, so the lower bounds for \(| L/pL+E_{p- 1}|\) obtained earlier by the same authors give nothing for \(| L/I| =| \bar B(2,p)|\). It's also worth to point out that some upper bound for \(| \bar B(2,p)|\) has also been obtained recently by \textit{S. I. Adyan} and \textit{A. A. Razborov} [Usp. Mat. Nauk 42, No.2, 3-68 (1987; Zbl 0627.17008)].
2-generator finite group of prime exponent
scientific article; zbMATH DE number 4095764