The growth of finite subgroups in p-groups
Alexander Yu. Olshanskii · Cambridge University Press eBooks · 1999
For an infinite p -group with a finite set of generators A , the value of the growth function (of the orders) of finite subgroups f(r) is defined to be the maximal size of a finite subgroup lying in the ball of radius r with respect to A . An equivalence class of the function f does not depend of the choice of a generating system. Growth rates of finite subgroups in p -groups, especially in 2-groups, are under investigation in the paper. Introduction As Schmidt knew, any infinite 2-group contains finite subgroups of unbounded order. On the other hand, by the Novikov-Adian theorem, in the free Burnside group of sufficiently large odd exponent n , every finite subgroup has order at most n . Moreover, it can happen that in an infinite p -group, where p is a sufficiently large prime number, all proper subgroups are cyclic of order p . Thus, the collection of orders of finite subgroups can has exhibit diverse behaviour in various periodic groups, and 2-groups are a special case. For a numerical estimate of the set of finite subgroups, we introduce an asymptotic invariant, namely, the growth of this set. (An analogue could be investigated in other algebraic systems). The case of 2-groups is considered below at greater length.