ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 23 Nov 2013 22:09:48 +0100pair grouphttps://ask.sagemath.org/question/10762/pair-group/Is there a way to implement the "pair group" in sage like there is in mathematica?
sf. http://mathworld.wolfram.com/PairGroup.html
Thu, 21 Nov 2013 20:36:52 +0100https://ask.sagemath.org/question/10762/pair-group/Comment by John Palmieri for <p>Is there a way to implement the "pair group" in sage like there is in mathematica?</p>
<p>sf. <a href="http://mathworld.wolfram.com/PairGroup.html">http://mathworld.wolfram.com/PairGrou...</a></p>
https://ask.sagemath.org/question/10762/pair-group/?comment=16647#post-id-16647I'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?Sat, 23 Nov 2013 11:11:15 +0100https://ask.sagemath.org/question/10762/pair-group/?comment=16647#post-id-16647Comment by John Palmieri for <p>Is there a way to implement the "pair group" in sage like there is in mathematica?</p>
<p>sf. <a href="http://mathworld.wolfram.com/PairGroup.html">http://mathworld.wolfram.com/PairGrou...</a></p>
https://ask.sagemath.org/question/10762/pair-group/?comment=16641#post-id-16641I 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.Sat, 23 Nov 2013 15:41:19 +0100https://ask.sagemath.org/question/10762/pair-group/?comment=16641#post-id-16641Comment by ClemFanJC07 for <p>Is there a way to implement the "pair group" in sage like there is in mathematica?</p>
<p>sf. <a href="http://mathworld.wolfram.com/PairGroup.html">http://mathworld.wolfram.com/PairGrou...</a></p>
https://ask.sagemath.org/question/10762/pair-group/?comment=16638#post-id-16638The pair group normally comes up in Polya enumeration of non-isomorphic graphs. So if we are looking at graphs with p vertices, then the group acting on the vertices is S_p. This induces a group on the edges of G, namely the pair group, S_p^{(2)}. I believe that Harary defined the pair group in "Graph Theory" and "Graphical Enumeration." I'm just curious if it is implemented in Sage the way it is in Mathematica. Sat, 23 Nov 2013 22:09:48 +0100https://ask.sagemath.org/question/10762/pair-group/?comment=16638#post-id-16638Comment by ClemFanJC07 for <p>Is there a way to implement the "pair group" in sage like there is in mathematica?</p>
<p>sf. <a href="http://mathworld.wolfram.com/PairGroup.html">http://mathworld.wolfram.com/PairGrou...</a></p>
https://ask.sagemath.org/question/10762/pair-group/?comment=16655#post-id-16655In this case, p is the number of letters in the symmetric group. So, S_p is the pth symmetric group.Fri, 22 Nov 2013 18:42:48 +0100https://ask.sagemath.org/question/10762/pair-group/?comment=16655#post-id-16655Comment by John Palmieri for <p>Is there a way to implement the "pair group" in sage like there is in mathematica?</p>
<p>sf. <a href="http://mathworld.wolfram.com/PairGroup.html">http://mathworld.wolfram.com/PairGrou...</a></p>
https://ask.sagemath.org/question/10762/pair-group/?comment=16656#post-id-16656Can you provide a definition of a pair group? (I looked at the link, but I don't know what "p" is.)Fri, 22 Nov 2013 15:01:09 +0100https://ask.sagemath.org/question/10762/pair-group/?comment=16656#post-id-16656Comment by ClemFanJC07 for <p>Is there a way to implement the "pair group" in sage like there is in mathematica?</p>
<p>sf. <a href="http://mathworld.wolfram.com/PairGroup.html">http://mathworld.wolfram.com/PairGrou...</a></p>
https://ask.sagemath.org/question/10762/pair-group/?comment=16642#post-id-16642In 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.Sat, 23 Nov 2013 14:34:09 +0100https://ask.sagemath.org/question/10762/pair-group/?comment=16642#post-id-16642