pair group

asked 2013-11-21 13:36:52 -0500

ClemFanJC07 gravatar image

updated 2015-01-14 03:56:25 -0500

FrédéricC gravatar image

Is there a way to implement the "pair group" in sage like there is in mathematica?

sf. http://mathworld.wolfram.com/PairGrou...

edit retag flag offensive close merge delete

Comments

Can you provide a definition of a pair group? (I looked at the link, but I don't know what "p" is.)

John Palmieri gravatar imageJohn Palmieri ( 2013-11-22 08:01:09 -0500 )edit

In this case, p is the number of letters in the symmetric group. So, S_p is the pth symmetric group.

ClemFanJC07 gravatar imageClemFanJC07 ( 2013-11-22 11:42:48 -0500 )edit

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?

John Palmieri gravatar imageJohn Palmieri ( 2013-11-23 04:11:15 -0500 )edit

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.

ClemFanJC07 gravatar imageClemFanJC07 ( 2013-11-23 07:34:09 -0500 )edit

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.

John Palmieri gravatar imageJohn Palmieri ( 2013-11-23 08:41:19 -0500 )edit