Let W = G ≀ H be the wreath product of G by an n-generator abelian group H. We prove that every element of W′ is a product of at most n+2 commutators, and every element of W2 is a product of at most 3n+4 squares in W. This generalizes our previous result.
Let G be a group and C = [G, G] be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. The exact values of c{G) are computed when G is a free nilpotent group or a free abelian-by-nilpotent group. If G is a free nilpotent group of rank n>2 and class c>2, c(G) = [n/2] if c = 2 and c(G) = n if c>2. If G is a free abelian-by-nilpotent group of rank n > 2 then c(G) = n.
Let G be a group and let M and N be two normal subgroups of G. Let [Formula: see text] be the set of all automorphisms of G which centralize G/M and N. In this paper, we find certain necessary and sufficient conditions on G such that [Formula: see text] be equal to Z( Inn (G)), Inn (G), C* or Aut c(G). We also characterize the subgroups of a finite p-group G for which the equality [Formula: see text] holds.
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