# Definition: Group

Definition:  group is a set $G$ with an associative (see below) function $g:G \times G\rightarrow G$, an element that we denote by $0$ such that for any $a\in G$, $g(a,0)=g(0,a)=a$, and such that for any $a\in G$, there exists an element that we denote by $a^{-1}$ such that $g(a,a^{-1})=g(a^{-1},a)=0$.//

Thus a group is nothing but a set with a way of combining things, an element (called the identity (we can use “the” as opposed to “an” because we prove that identities are unique in lesson 23)) that doesn’t change things when it’s combined with them, and a way of “getting back to the identity” from any element (inverses).

By associative, we mean that for any $a,b,c\in G$, we have $g(g(a,b),c)=g(a, g(b,c))$.  This simply means that “combining the first two, and then the third” is the same as “combining the last two, and then the first”.  It most certainly does not necessarily mean that $g(a,b)=g(b,a)$, so be warned.  For a more detailed introduction to groups, see lesson 22.

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