How to write small size text between the itemised?

I want to write between the itemise, that is after one bullet point, I want to add 2 small sentences below it. See my code and what I want is given below:

\documentclass{amsart}
\newtheorem{thm}{Theorem}
\usepackage{xcolor}

\begin{document}
\section{\textcolor{blue}{Testing Nilpotence in linear time}}
Given a group $G$ in the form of mutiplication table, we want to check to decide  divides the order of $G$}. Algorithm for testing nilpotence is given below. \\

\begin{itemize}
\item Compute the prime factorization of $n= p_1^{\alpha_1} \times p_2^{\alpha_2} \cdots p_i^{\alpha_i}$. \\
\item Determine the order of all elements in $G$. \\
\item For $1 \le i \le r$, check if $\mathcal{N}(p_i^{\alpha_i}) \neq p_i^{\alpha_i}$ then $G$ is not nilpotent. \\
\item Else output that $G$ is nilpotent
\end{itemize}

see compiled output : Question : How to add small text as given in the image? I suggest you create a little utility macro called \aside. If \footnotesize makes the text too small for your taste, use \small instead. \documentclass{amsart}
\usepackage{xcolor}
\newcommand\aside{\par\quad{\footnotesize(#1)}\par} % or "\small", if you prefer

\begin{document}
\section{\color{blue}Testing Nilpotence in linear time}

Given a group $G$ in the form of multiplication table, we want to check to
decide [...] divides the order of $G$. An algorithm for testing nilpotence
is given below.

\begin{itemize}
\item Compute the prime factorization of $n = p_1^{\alpha_1} \times p_2^{\alpha_2} \cdots \times p_i^{\alpha_i}$.
\aside{Each $p_i^{\alpha_i}$ is the highest power}

\item Determine the order of all elements in $G$.
\aside{It can be done easily}

\item For $1 \le i \le r$, check if $\mathcal{N}(p_i^{\alpha_i}) \neq p_i^{\alpha_i}$. If true, $G$ is not nilpotent.

\item Else, conclude that $G$ is nilpotent.
\end{itemize}

\end{document}
• your mwe is not complete (missing is \end{document})
• it also contain error (superfluous } after $G$)
• all \\ after items are superfluous
• additional line below (main) item just write in new line (you can select smaller font size for this line(s) )
• for better formatting use package enumitem

\documentclass{amsart}
\newtheorem{thm}{Theorem}
\usepackage{xcolor}
\usepackage{enumitem}

\begin{document}
\section{\textcolor{blue}{Testing Nilpotence in linear time}}
Given a group $G$ in the form of multiplication table, we want to check to decide  divides the order of $G$. Algorithm for testing nilpotence is given below. %\\ had to be removed

\begin{itemize}[itemsep=1ex,leftmargin=1cm]
\item Compute the prime factorization of $n= p_1^{\alpha_1} \times p_2^{\alpha_2} \cdots p_i^{\alpha_i}$.               % "\\" had to be removed

{\small(small text in the next line)}
\item Determine the order of all elements in $G$. % "\\" had to be removed

{\small(small text in the next line)}
\item For $1 \le i \le r$, check if $\mathcal{N}(p_i^{\alpha_i}) \neq p_i^{\alpha_i}$ then $G$ is not nilpotent.            % "\\" had to be removed

{\small(small text in the next line)}
\item Else output that $G$ is nilpotent
\end{itemize}
\end{document} 