Cryptic definition of power group explanation I'm given the following cryptic definition of power group. Please explain me its details. 
Given a group $(G,*)$ and a set $I$ we consider $G^I$ (which I understand that is the set of all mappings from $I$ to $G$) and $(a_i)_{i \in I} \times (b_i)_{i \in I} = (a_ib_i)_{i \in I}$. This is surprisingly how it ends. The text gives also an example.
$\mathbb{Z}^{a,b,c}$ is the set of mappings from $\{a,b,c\}$ to $\mathbb{Z}$ with the sum $f+g$ defined as $(f+g)(x) = f(x) + g(x)$.
To be precise what I need to understand is:


*

*What does the expression $(a_i)_{i \in I} \times (b_i)_{i \in I} = (a_ib_i)_{i \in I}$ mean? 

*Is the power group a group. If so with what operation. 
 A: Perhaps it's confusing you that the definition uses indexed-family notation rather than function notation? We could also write it as:

Let $(G,*)$ be a group and let $I$ be any set. The power group $G^I$ is then then the group whose elements are all possible functions $I\to G$, and where the group operation $\bar*$ is defined such that the product of $f$ and $g$ is the function $f\bar *g$ defined by
  $$ (f\bar * g)(i) = f(i) * g(i) \qquad \text{ for all }i\in I$$

A: It appears to me from the example that $G^I$ consists of elements of $G$ enumerated by elements from $I$. An element $(a_i)_{i\in I}$ of $G^I$ is recognized as an enumerated list of elements $a_i\in G$ which in turn may be identified with the map $i\mapsto a_i$ from $I$ to $G$.
The relation
$$
(a_i)_{i\in I}\times(b_i)_{i\in I}=(a_ib_i)_{i\in I}
$$
means that elements of $G^I$ are composed by composing index-wise the elements of each as elements of $G$. So $(a_i)_{i\in I}$ and $(b_i)_{i\in I}$ both from $G^I$ will be composed by composing $a_i$ and $b_i$ in $G$ for each index $i\in I$. Thus $i\mapsto a_i$ and $i\mapsto b_i$ composed becomes $i\mapsto a_ib_i$.
Finally, this gives us a group with $(e)_{i\in I}$ understood as the map $i\mapsto e$ as the neutral element in $G^I$, where $e$ denotes the neutral element of $G$.

BTW, note that the number of elements in $G^I$ must be $|G|^{|I|}$ whenever $G$ and $I$ are both finite.
