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A group is a monoid $(G, *)$ (i.e. $*$ is associative and has an identity element $e$) such that for every element $x$ element $G$ there exists an element $y$ element $G$((i.e $\forall x\in G \exists y\in G$)) such that $xy = yx = e$. Such an element $y$ is called a symmetrical element or an inverse (element) for $x$.

and the second definition is: A set with an internal operation $*$ is a group if and only if

$(i)$ is associative,

$(ii)$ given any two elements $x, y$ element of $G$, there exist unique elements $a, b$ element of $G$ such that $ax = y$ and $xb = y$.

and that the definition of equivalent is that it has to satisfy the following three axioms. $x R x$, $x R y$ implies $y R x$ and $x R y$ implies that if $y R z$ then $x R z$

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    An easier way of showing equivalence between the two definitions is to show that if an object is a group under definition 1 then it is a group under definition 2, and also that if it is a group under definition 2 it is a group under definition 1. – 2017-02-04
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    The definition of "equivalent" you looked up applies to a completely different context (the one of "equivalence relation on a set"). Here, it just means that a pair set-binary operation $(G,*)$ satisfies condition $(a)$ if and only if it satisfies condition $(b)$. – 2017-02-04

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