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How to show $\langle x, y\rangle=S_7$? For arbitrary $2$ cycle $x$ and $7$ cycle $y$ in $S_7$?

So far I have by Lagrange's that $2 | \langle x,y\rangle$ and $7|\langle x,y\rangle$ and $\langle x,y\rangle|S_7$.

In other questions of a similar nature I have had success by doing case by case and ruling out all possibilities but here since $|S_7|=7!=5040$ I don't see that as being very helpful.

Any ideas?

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  • $\begingroup$ This is related: math.stackexchange.com/a/357673/160300 $\endgroup$
    – Crostul
    Commented May 10, 2017 at 11:15
  • $\begingroup$ Can you tell us what your angle-brackets mean? If the question, it appears that they represent some group or subgroup (else they could not equal $S_7$). In the text, you have $2$ dividing an angle-bracket thing, suggesting it's an integer. Clarify, please? $\endgroup$ Commented May 10, 2017 at 11:15
  • $\begingroup$ I guess it's the generating set of the group $S_7$ $\endgroup$
    – Arpan1729
    Commented May 10, 2017 at 11:34

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$(1,2)$ and $(1,2,...,n)$ generate $S_n$. Conclude from this that $(i,j)$ and $(1,2,...,n)$ generate $S_n$ whenever $gcd(i-j,n)=1$. Your problem is solved.

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