Approximate centralizer

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Definition

Definition with symbols

Let $G$ be a group and $g\in G$ be an element. Then, the approximate centralizer of <amth>g</math> in $G$, denoted as $C_{G}^{*}(g)$, is defined as the union of centralizers of all positive (or equivalently, all nonzero) powers of $g$.

The approximate centralizer is always a subgroup, and it contains the centralizer. When $g$ has finite order, its approximate centralizer is the whole group. Hence, the approximate centralizer makes sense, or is worth studying, only for elements of infinite order.

References

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