A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954, B. H. Neumann discovered that if G is a BFC-group then the derived group G' is finite. Let w = w(x(1), ..., x(n)) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if vertical bar x(G)vertical bar <= m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If vertical bar x(w(G))vertical bar <= m for every w-value x, then [w(w(G)), w(G)] is finite of order bounded by a function of m and n.
Detomi, E., Morigi, M., Shumyatsky, P. (2019). BFC-theorems for higher commutator subgroups. QUARTERLY JOURNAL OF MATHEMATICS, 70(3), 849-858 [10.1093/qmath/hay068].
BFC-theorems for higher commutator subgroups
Morigi, Marta;
2019
Abstract
A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954, B. H. Neumann discovered that if G is a BFC-group then the derived group G' is finite. Let w = w(x(1), ..., x(n)) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if vertical bar x(G)vertical bar <= m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If vertical bar x(w(G))vertical bar <= m for every w-value x, then [w(w(G)), w(G)] is finite of order bounded by a function of m and n.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
