Upper bounds in the restricted Burnside problem
The question of whether the orders of all finite \(m\) generator groups of exponent \(n\) are bounded above by a function \(f(m,n)\) of \(m\) and \(n\), known as the restricted Burnside problem, has an affirmative answer. The authors give a better upper bound for the orders of \(m\) generator groups of prime and prime power exponent. For each \(n\geq 0\), define classes of Grzegorchyk hierarchy as follows: \(\text{Gr}_ 0 (x,y)= x+y\), \(\text{Gr}_ 1 (x)= x^ 2+2\), \(\text{Gr}_{n+2} (0)=2\), \(\text{Gr}_{n+2} (x+1)= \text{Gr}_{n+1} (\text{Gr}_{n+2} (x))\). The class \(\text{Gr}^ n\), \(n\geq 1\), consists of functions which can be obtained from the zero function, the successor function, the projection functions, and the function \(\text{Gr}_{n-1}\) by composition and limited recursion. Let \(T(m,1)=m\) and \(T(m, n+1)= m^{T(m, n)}\). The authors prove that if \(G\) is a finite \(m\) generator group, \(m>1\), of exponent \(q= p^ k\), where \(p\) is a prime, then \(| G|\leq T(m, q^{q^ q})\) and this upper bound lies in \(\text{Gr}^ 4\), whereas the previously obtained upper bounds were of class \(\text{Gr}^ 5\). They also show that if \(G\) is a finite \(m\) generator group of prime exponent \(p\geq 29\) then \(| G|\leq T(m, 2^ p)\) and for \(p<29\), \(| G|\leq T(m, 3^ p)\).
finitely generated groups of prime-power exponent
finite \(m\) generator groups of exponent \(n\)
scientific article; zbMATH DE number 554200