Some interconnections between group theory and logic
T. Tollis · Cambridge University Press eBooks · 1987
In his classic paper [5] Mal'cev applied certain methods of mathematical logic in order to prove local theorems for various classes of groups. In [1] Cleave simplified certain ideas of Mal'cev and proved the local theorems for the same classes of groups. In this paper we describe a metamathematical method similar to that of Mal'cev and Cleave. Using this method we study properties of various classes of finite and infinite groups expressed by second order sentences which will be called strongly boolean- (universal-existential). We say that a class of groups 3 has the B -property if G 1 , G 2 € J and G 1 < G < G 2 imply G € J, (B-property is the abbreviation of Betweenness property, meaning that whenever a group G is “between” two groups which belong to a class then necessarily G is in the class. As usual H