How to write elements of uncountable products of a group?

Question

Suppose $H$ is a group and I take $G=H^{I}$ i.e. uncountable product of $H$ with itself. Now how will I denote an element of $G$, I cant do it in terms of tuple is I know. Suppose i want to write an element where only one of the elements is non identity, how will I denote it?

Answer 1

The product $H^I$ is the set of functions $I \to H$. So, suppose we have an element in this product that is the identity everywhere except on one place. Say the value is $h\neq id$ on place $i\in I$. Then we can write this element which I call $f$ as

$$f(j)= \begin{cases} id \quad j\neq i\\ h \quad j=i\end{cases}$$