If G is a group, two elements x, y of \(G\setminus Z(G)\) are said to have finite distance \(d(x,y)=d+1\) if d is the least non-negative integer such that there exist elements \(u_ 1,...,u_ d\) in \(G\setminus Z(G)\) for which \(xu_ 1=u_ 1x\), \(u_ 1u_ 2=u_ 2u_ 1,...,u_ dy=yu_ d\); if such elements do not exist then x and y have distance \(\infty\). A group G has finite distance m if m is the least positive integer such that d(x,y)\(\leq m\) for each pair (x,y) of elements of \(G\setminus Z(G)\) with finite distance. An N-complex of a group G is either the centre Z(G) of G or a subset N of G such that, if \(g\in N\setminus Z(G)\), then N contains every element h of G for which \(hg=gh.\)
It is proved that a free group G of rank \(\geq 2\) has distance \(d(G)=1\), the minimal N-complexes of G are precisely the maximal cyclic subgroups, and G is the set-theoretic union of them (and two different maximal cyclic subgroups have trivial intersection). Moreover the minimal N- complexes are infinite, self-normalizing and maximal nilpotent subgroups of G.
Also it is proved that the minimal N-complexes of a Burnside group B(n,e) \((n>1\), \(e\geq 665\), e odd) are the cyclic subgroups of order e, and they are precisely the maximal nilpotent subgroups (any two of them have trivial intersection); such groups have distance \(d(G)=1\) and are the set-theoretic union of their minimal N-complexes. Similar results are proved also for the group A(n,e) \((n>1\), \(e\geq 665\), e odd) (for the definition of A(n,e) see [S. I. Adyan, The Burnside Problem and Identities in Groups (1979; Zbl 0417.20001)]).