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Given the permutation (1, 2, 4)(3, 5, 6) is it clear which symmetric group this permutation belongs to? Explain.

so from here I got:

1 2 3 4 5 6

2 4 5 1 6 3

and that is an element of S6

Im not exactly sure how to find the symmetric group this permutation belongs to, or if the matrix above is the symmetric group?

or would it be: {2,4,5,1,6,3}???

And yes I know I need to learn latex, I will. but I do not have time to learn it before my final.

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  • $\begingroup$ It belongs to $S_n$ where $n$ is the number of different symbols in your permutation. $\endgroup$ Dec 17, 2014 at 7:29

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The permutation is a permutation of $\{1,2,\ldots,6\}$.

The symmetric group $S_6$ is defined as the set of permutations of $\{1,2,\ldots,6\}$ (with the group operation: composition of permutations).


One caveat: the permutation is in cycle notation, and it's possible that some fixed points are omitted from the notation. E.g., it might be shorthand for $(1,2,4)(3,5,6)(7)$. In this case, it's not clear. (It is typical for authors avoid ambiguity in these situations by explicitly stating which symmetric group they're working in.)

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  • $\begingroup$ However one can always interpret $S_n \subseteq S_m$ for all $n \leq m$, so the caveat should not be that much of a problem. $\endgroup$
    – j4GGy
    Dec 17, 2014 at 8:08

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