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What is the concise definition of a relation, regarding the presentation of a group through generators and relations? what is a relation?

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    Every group $G$ is (isomorphic to a) quotient of a free group $F(X)$ on some set $X$ (take $X$ to be the generators of $G$). Thus, $F(X)/N \cong G$, then the homomorphic image of elements of $N$ are called relations on $G$. – 2017-01-17
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    @user2902293 I don't understand what you mean by the homomorphic image of elements of $N$. The elements of $N$ are usually called relators rather than relations. A relation of $G$ is an expression $w_1=w_2$ with $w_1,w_2 \in F(X)$ and $w_1N=w_2N$. – 2017-01-17
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    All elements in the kernel are mapped to the group identity. Relations is a subset which tracks what you need to mod out (in the free group) to get the group G. – 2017-01-17

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A relation is an equation that holds for the generators. For instance if one of the generators $a$ is of order 2 then $a^2 = e$ is a relation. If generators $b$ and $c$ commute then $bcb^{-1}c^{-1} = e$ is a relation.

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    Relations: equations of the generating elements that equal the identity element??? – 2017-01-17
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    @ParthVader They don't really need to equal the identity element. Just any equation amongst the generators will do. It's just that usually people rearrange the relation so that $e$ is on one side, just like people often rearrange equations so that 0 is on one side. For instance, my second example relation probably looks more familiar if you write it $bc =cb.$ – 2017-01-17
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Typically, a relation is a combination of elements whose product (within the group) gives the identity.

For example, the dihedral group with $2N$ elements has presentation $$ \langle x,y : x^2 = 1, y^N = 1, (xy)^2 = 1 \rangle.$$ Sometimes the statement that they are all equal to the identity is shortened, or perhaps removed, leading to the two alternate ways of giving the presentation $$ \langle x,y: x^2 = y^N = (xy)^2 = 1\rangle, \qquad \langle x,y : x^2, y^N, (xy)^2 \rangle.$$

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    Relations: equations of the generating elements that equal the identity element is that correct – 2017-01-17