Department of Mathematics

A generalized torsion element in a group \( G \) is a nontrivial element \( g \in G\), such that \(h_1^{-1}gh_1\dotsm h_n^{-1}gh_n=1 \) for some \( h_i \in G\) ; i.e, a product of conjugates of \( g \) is trivial. (This is a natural generalization of torsion elements—elements of finite order—as suggested by its name.)

As an example of application of our main theorem, we know now—it was not known before—that the following group doesn't a have generalized torsion element:
The group \( G \) given by generators \( a,\,b,\,a',\,b' \) and relations \[ a^3b^3a^5b^5=a'^3b'^3a'^5b'^5,\, a^7b^{-7}a^{11}b^{-11}=a'^7b'^{-7}a'^{11}b'^{-11}\,. \]

We will talk about

1. Some examples of groups without generalized torsion elements
2. Why is it interesting to study generalized torsion and amalgams of groups
3. Our results in comparison with others.

This is a joint work with Adam Clay.