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I'm studying group theory, and recently i read in a part that if we have a presentation $$\langle S | R \rangle$$ in wich in the right side we have an equation like $x=y$ that mean that we have a presentation in wich $y^{-1}x∊R$ That is true? if yes why $y^{-1}x∊R$ is the same that in the presented group $x=y$?

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If you have a semigroup presentation (so you are not given an identity or inverses) then $R$ must consist of things of the form $x=y$. – user1729 May 29 '12 at 9:08

You might see $y^{-1}x$ referred to as a ''relator'' while $y=x$ may be referred to as a ''relation.'' They both give you the same information though.

The point of $R$ is to fully describe the group, since in general, you need to know more than just how many generators there are. For example $\langle a \rangle$ is the infinite cyclic group (no nontrivial relations) but $\langle a\;|\; a^n=1 \rangle = \langle a\;|\; a^n \rangle$ is the finite cyclic group of order $n$.

For more information, see Section 40 in Fraleigh's "A First Course in Abstract Algebra." It's a very gentle introduction to the theory of group presentations.

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