pair group
Is there a way to implement the "pair group" in sage like there is in mathematica?
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Can you provide a definition of a pair group? (I looked at the link, but I don't know what "p" is.)
In this case, p is the number of letters in the symmetric group. So, S_p is the pth symmetric group.
I'm reading the first sentence of Wolfram's definition: if `G` is a group, what's `p`? Or does the pair group depend on both `G` and `p` independently? Or does `G` come with an embedding into a specific symmetric group?
In this case, G=S_p. In any case, if G is finite, then it is (isomorphic to) a subgroup of S_p for some suitably chosen p (Cayley's Theorem). The pair group only depends on G. However, the interesting and important case is when we use the full symmetric group for G.
I know Cayley's theorem, but my point is that there is no unique `p` so that `G` is a subgroup of `S_p`. (Plus the cited definition doesn't even insist that the group is finite. Their first sentence is extremely sloppy; I wouldn't let my algebra students get away with it.) So the precise description of the pair group (in terms of which permutations in contains) will definitely depend on the choice of embedding of `G` into `S_p`. Different embeddings should lead to different, but (I'm willing to believe) isomorphic, groups.