# Definition Consider two [[Group|groups]] $G_1$ and $G_2$ with binary operations $\bullet$ and $*$ respectively. They are called *isomorphic* (i.e. identical in their [[group]] properties) if there is an [[Types of Maps|bijection]] from $G_1$ to $G_2$ which respects the [[group]] operations. For $g, h \in G_1$ and $f:G_1 \rightarrow G_2$, then $ f(g \bullet h) = f(g) * f(h) $ We call $f$ a *group isomorphism*. If the two groups are isomorphic, we write: $ (G_1, \bullet) \cong (G_2, *) $ In simpler language, the two groups are isomorphic if there exists a *one-to-one* correspondence between their [[Group Element|elements]]: $ \begin{align} a \rightarrow \hat{a},\\ b \rightarrow \hat{b},\\ ab \rightarrow \hat{a}\hat{b},\\ \end{align} $ where the "unhatted" letters are elements of the first group and the hatted ones are elements of the second group. If the correspondence is *many-to-one*, then it is a [[Group Homomorphism|homomorphism]]. Note that, since isomorphisms are bijections, they are invertible and thus set up a one-to-one correspondence that fully preserves the group structure. Thus, if two groups $G$ and $H$ are isomorphic, then we can think of them as the same group but with different labels for the [[group element|group elements]]. # Examples Consider the permutation group $P(3)$ and the symmetry operations of the equilateral triangle. It is well known that there is a one-to-one correspondence between these two groups, so we say these groups are isomorphic.