Approximate normalizer

Definition with symbols
Let $$G$$ be a group and $$g \in G$$ be an element. Then, the approximate normalizer of $$g$$, denoted as $$N_G^*(g)$$ is defined as the set of all elements $$t \in G$$ for which there exist nonzero integers $$m,n$$ such that $$t^{-1}g^mt = g^n$$.

The approximate normalizer of any element is a subgroup of the whole group. It equals the whole group if the element has finite order (viz, is a torsion element). Thus, the notion makes sense to study only for elements of infinite order.