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The question reads as follows:

Suppose that $H$ is a subgroup of $A_6$ with index $|A_6: H| \leq 4$. By considering the cosets $H\sigma ^i$, show that any 5-cycle $\sigma$ belongs to $H$. Hence, show that $H = A_6$.

I know that through using Lagrange's theorem the order of $H$ is greater than or equal to 90, and that $A_6$ is a simple group. But I'm not sure how to use the cosets to verify containment.

Desperado
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Bond
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Apr 18 '23 at 16:40
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    How can a $5$-cycle act on $4$ cosets? – Derek Holt Apr 18 '23 at 16:44
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    Here's a MathJax tutorial :) – Shaun Apr 18 '23 at 16:54
  • @DerekHolt is this a criticism of the question or a hint? – Bond Apr 18 '23 at 16:57
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    A hint! The group acts by multiplication on the four cosets of $H$ in $G$. – Derek Holt Apr 18 '23 at 17:13
  • yes, so let x be a 5 cycle not in H, then consider H, xH, ... . then since this is a group of order less than or equal to 4, at least two of the cosets are equal. But this is not possible and hence contradiction? – Bond Apr 18 '23 at 17:27

1 Answers1

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As you see, two of the cosets $H\sigma^i\ (i=0,\dots,4)$ for a 5-cycle $\sigma$ must coincide since $|A_6\colon H|\leq 4$. Namely $H\sigma^i=H\sigma^j$ for some $i<j<5$ and $H\sigma^{j-i}=H$, $\sigma^{j-i}\in H$, hence $\sigma\in H$ follows.

Thus, $H$ contains all 144 5-cycles in $A_6$ and so $H$ has 36 5-Sylow subgroups. By index assumption, $H$ is of order 360, 180, 120 or 90. The latter two are not multiples of 36 so impossible. The second case is where $H$ is of index two and so normal in $A_6$, thus impossible by simplicity. Therefore $H=A_6$.

Ayaka
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