Tall curly brace

The following may be close to what you're looking for:

\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{gather*}
c \colon \{1, \dots, n\} \rightarrow \{1, \dots, n\} \text{ such that}\\    
\begin{cases}
  c(a_i) = a_{i+1} & \text{for }1\le i<l\\    
  c(a_l) = a_1    
\end{cases}
\end{gather*}
\end{document}

Some comments:

• Use the cases environment, provided by the amsmath package, to place a left curly brace ahead of two or more expressions.
• Don't use ... to create a typographical ellipsis; instead, use \dots.
• Use the \text{ } command to write text material inside mathematical expressions.
• As pointed out in the comments by @SvendTveskæg and @daleif, it's better (typographically speaking) to use the command \colon instead of :.

---

On the other hand, if you need the left-hand curly brace to span all three rows, you could do so with the following code:

\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
\begin{cases}
  c \colon \{1, \dots, n\} \rightarrow \{1, \dots, n\} \text{ such that}\\      
  c(a_i) = a_{i+1}  \text{ for $1\le i<l$}\\
  c(a_l) = a_1
\end{cases}
\end{equation*}
\end{document}

Observe, though, that this code doesn't make full use of the capabilities of the cases environment; for sure, one could replace \begin{cases} ... \end{cases} with a more basic \left\{\begin{array}{@{}l} ... \end{array}\right. construct.

Although the cases environment probably suits your needs best, I'd like to mention the more versatile and (probably) equivalent syntax \left\{ <blank> \right\}for future use, in case you haven't come across it.