Approximate normalizer

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This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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^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.

References

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