Is H a normal subgroup in G?

Question

Let $G = S_5$ and let $H = \langle(1, 2, 3, 4, 5)\rangle$. Is $H$ a normal subgroup of $G$ ?

Having some trouble figuring out this problem, it would be great if someone can help to find it!

Answer 1

Consider $(12)(12345)(12)=(13452)\not\in\langle (12345)\rangle =\{e,(12345),(13524),(14253),(15432)\}$.

Answer 2

Chris Custer provides an answer by using definition but I hope the following is useful to you for finding normal subgroups in $S_n$

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Hint: A subgroup $H$ of $G$ is normal $\iff$ $H$ is the union of conjugacy classes $$\&$$ Remember that, elements of $S_n$ are conjugate $\iff$ they have the same cycle type