TL;DR
This survey provides a comprehensive overview of perfect isometries, including proofs of existing results and new observations, to clarify and advance understanding of this mathematical concept.
Contribution
It compiles and proves key results on perfect isometries and introduces new insights not previously documented.
Findings
Collected and proved numerous results on perfect isometries.
Presented new observations enhancing understanding of the concept.
Clarified the foundational aspects of perfect isometries.
Abstract
This paper is an introduction and a survey to the concept of perfect isometries which was first introduced by Michel Brou{\'e} in 1990. Our main aim is to provide proofs of numerous results scattered in the literature. On the other hand, we make some observations which did not appear anywhere before.
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