Computations in symmetric groups

#group_theory

Let $S_{n}$ denote the symmetric group on $n$ letters.
(a) Is the element $(1 2 3 4) (2 5 3 4 6) (1 5 3 2 4 7) \in S_{7}$ even or odd? Indicate your reasoning.
(b) Find the order of $(1 3 4) (2 4 3) (1 3 4) \in S_{4}$ . Show all work.
(c) Write $(1 5 2 3) (2 1 3 4) (1 5 2 3)^{- 1} \in S_{5}$ in disjoint cycle form. Show all work.

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Let $S_n$ denote the symmetric group on $n$ letters.
\begin{enumerate}[label=\alph*)]
	\item Is the element $(1\,2\,3\,4)(2\,5\,3\,4\,6)(1\,5\,3\,2\,4\,7)\in S_7$ even or odd? Indicate your reasoning.
	\item Find the order of $(1\,3\,4)(2\,4\,3)(1\,3\,4)\in S_4$. Show all work.
	\item Write $(1\,5\,2\,3)(2\,1\,3\,4)(1\,5\,2\,3)^{-1}\in S_5$ in disjoint cycle form. Show all work.
\end{enumerate}