Prove that \(A_{n}\) is simple for \(n \geq 5\), following the steps and hints
given.
a. Show \(A_{\text {e contains cuery } 3 \text {-cycle if } n \geq 3 \text {.
}}\)
b. Show \(A_{n}\) is generated by the 3-cycles for \(n \geq 3\). [Hint: Note that
\((a, b)(c, d)=(a, c, b)(a, c, d)\) and \((a, c)(a, b)=(a, b, c) .]\)
c. Let \(r\) and \(s\) be fixed elements of \(\\{1,2, \cdots, n]\) for \(n \geq 3\).
Show that \(A_{n}\) is generated by the \(n\) "special" 3-cycles of the form \((r,
s, i)\) for \(1 \leq i \leq n\) [Hint: Show every 3-cycle is the product of
"special" 3-cycles by computing
$$
(r, s, i)^{2}, \quad(r, s, f)(r, s, i)^{2}, \quad(r . s, j)^{2}(r, s, i),
$$
and
$$
(r, s . i)^{2}(r, s, k)(r, s, j)^{2}(r, s, i)
$$
Observe that these products give all possible types of 3 -cycles.]
d. Let \(N\) be a normal subgroup of \(A_{n}\) for \(n \geq 3\). Show that if \(N\)
contains a 3-cycle, then \(N=A_{n}\). [Hint: Show that \((r, s, i) \in N\) implies
that \((r, s, j) \in N\) for \(j=1,2, \cdots, n\) by computing
$$
\left.((r, s)(i, j))(r, s, i)^{2}((r, s)(i, j))^{-1}\right]
$$
e. Let \(N\) be a nontrivial normal subgroup of \(A_{n}\) for \(n \geq 5\). Show
that one of the following cases must hold, and conclude in each case that
\(N=A_{n}\).
Case I \(N\) contains a 3-cycle.
Case II \(N\) contains a product of disjoint cycles, at least one of which has
length greater than 3. [ Hint: Suppose \(N\) contains the disjoint product
\(\sigma=\mu\left(a_{1}, a_{2}, \cdots, a_{n}\right)\). Show
\(\sigma^{-1}\left(a_{1}, a_{2}, a_{3}\right) \sigma\left(a_{1}, a_{2},
a_{3}\right)^{-1}\) is in \(N\). and compute it.]
Case III \(N\) contains a disjoint product of the form \(\sigma=\mu\left(a_{4},
a_{5}, a_{6}\right)\left(a_{1}, a_{2}, a_{3}\right)\). [Hint: Show
\(\sigma^{-1}\left(a_{1}, a_{2}, a_{4}\right)\) \(\sigma\left(a_{1}, a_{2},
a_{4}\right)^{-1}\) is in \(N\), and compute it.\\}
Case IV \(N\) contains a disjoint product of the form \(\sigma=\mu\left(a_{1},
a_{2}, a_{3}\right)\) where \(\mu\) is a product of disjoint 2 -cycles. [Hint:
Show \(\sigma^{2} \in N\) and compute it.]
Case \(\mathbf{V} N\) contains a disjoint product \(\sigma\) of the form
\(\sigma=\mu\left(a_{3}, a_{4}\right)\left(a_{1}, a_{2}\right)\), where \(\mu\) is
a product of an cven number of disjoint 2-cycles. [Hint: Show that
\(\sigma^{-1}\left(a_{1}, a_{2}, a_{3}\right) \sigma\left(a_{1}, a_{2},
a_{3}\right)^{-1}\) is in \(N\), and compute, it to deduce that
\(\alpha=\left(a_{2}, a_{4}\right)\left(a_{1}, a_{3}\right)\) is in \(N\). Using
\(n \geq 5\) for the first time, find \(i \neq a_{1}, a_{2}, a_{3}, a_{3}\) in
\(\\{1,2, \ldots, n]\). Let \(\beta=\left(a_{1}, a_{1}, i\right)\). Show that
\(\beta^{-1} \alpha \beta \alpha \in N\), and compute it.]
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