On the definition of product of groups What I'm asking comes from Bosch, Algebra, the first chapter on elementary group theory.
1) Let $X$ be a set and $G$ a group. Then the set $G^X$ of maps from $X$ to $G$ is a group in a natural way, that is for every $f,g\in G^X$, set $(f\cdot g)(x):=f(x)\cdot g(x)$.
2) Let $X$ be a set of indexes and $\left(G_x\right)_{x\in X}$ a collection of groups. Then the product set $\prod_{x\in X}G_x$ becomes a group if we define the product component-wise, that is $(g_x)_{x\in X}\cdot (h_x)_{x\in X}:=(g_x\cdot h_x)_{x\in X}$.
Now suppose that the groups $G_x$ are copies of the same group $G$. Then, using notations in 1), we have
$$\displaystyle\prod_{x\in X}G_x=G^X$$
Well, I can't understand this equation. I mean, an element in the lefthandside is a collection $(g_x)_{x\in X}$ of elements of $G$, whereas on the righthandside we have functions taking values in $G$. How can the two objects be the same?

Well, the indexing is simply a function on $X,$ yes? That is, $g=(g_x)_{x\in X},$ where $g_x\in G_x=G$ for all $x\in X,$ so in fact, $g$ is in effect a function $X\to G$ given by $g(x)=g_x$.
More generally, if we have a collection of sets $\{A_x\}_{x\in X},$ then $\prod\limits_{x\in X}A_x$ can be thought of as the set of all functions $f:X\to\bigcup\limits_{x\in X}A_x$ such that $f(x)\in A_x$ for all $x\in X$.
