Approximate normalizer

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Definition

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 $t^{-1}g^{m}t=g^{n}$.

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.

References

• On a certain infinite permutation group by Graham Higman, J. Algebra 131 (1990), 359-369