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I will start with example. I have two permutations (of 5 elements) - (1 2) and (1 2)(3 5 4). When the first permutation changes something, the second do the same change (but it can do anything with fixed elements of the first).

What's the best way (terminology) to call or understand such relation between two permutations?

[EDIT] Here is how to check for such relation in GAP:

ForAll(Orbits(Group(p1)), i -> i in Orbits(Group(p2)));

If true, then p2 contains all cycles of p1.

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Call the first one $\sigma$ and the second $\tau$. What you’re saying is that every cycle of $\sigma$ is a cycle of $\tau$. –  Brian M. Scott Feb 7 '13 at 23:25
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up vote 2 down vote accepted

I guess you're saying that the first permutation is the restriction of the second one to a subset left invariant by the second one.

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Did you mean "restriction of the second one to a subset left invariant by the first one"? Or something like that - because the second permutation in my example doesn't have any invariant subset, imho. –  zaarcis Feb 10 '13 at 5:18
    
@IlmārsCīrulis, the second permutation $\tau$ leaves exactly the two sets $\{1,2\}$ and $\{3,4,5\}$ invariant, the orbits of $\langle \tau \rangle$. In your case, the first permutation $\sigma$ is the restriction of $\tau$ to $\{1,2\}$. –  Andreas Caranti Feb 10 '13 at 6:44
    
Oh, a different kind of invariant. My apologies. Can you give context where such terminology is used? My original context was thinking about graph symmetries (automorphisms), where I thought that symmetry (1 2)(2 5 4) "includes" symmetry (1 2). But I didn't find anything like that. (And nothing about permutations too.) –  zaarcis Feb 10 '13 at 10:10
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