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On a combinatorial problem in group theory. (English) Zbl 0794.20041
The class \(DS(m)\), where \(m\) is an integer greater than 1, is defined as the class of the groups in which all \(m\)-sets have deficient squares, that is to say, in which for each set \(X\) of \(m\) elements the set \(X^ 2= \{xy\mid x,y\in X\}\) has fewer than \(m^ 2\) elements. The class \(DS\) is the union of the classes \(DS(m)\) for \(m\) ranging over all the integers \(>1\). The authors determine this class \(DS\) completely: it consists of just those groups in which either the subgroup generated by all squares of elements is finite, or there is an abelian subgroup of finite index on which each group element acts by conjugation either as the identity automorphism or as the inverting automorphism [such groups are called “nearly-dihedral” by the authors].
The proof makes use of a theorem of Peter M. Neumann, with his permission here first published with his proof; it says that all groups in \(DS\) are finite-by-abelian-by-finite. A corollary of the proof of this theorem is that all FC-groups [that is groups with finite classes of conjugate elements] in \(DS\) are BFC-groups [that is groups with boundedly finite classes of conjugate elements], or equivalently finite-by-abelian. The authors’ proof of their theorem requires distinguishing between FC-groups in \(DS\) and those that are not FC-groups. A final section shows by straightforward examples that the various finite quantities implicit in the authors’ theorem are not bounded.

MSC:
20E34 General structure theorems for groups
20E07 Subgroup theorems; subgroup growth
20F24 FC-groups and their generalizations
20F05 Generators, relations, and presentations of groups
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