Working with group cosets in MAGMA - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:42:21Z http://mathoverflow.net/feeds/question/92891 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92891/working-with-group-cosets-in-magma Working with group cosets in MAGMA dward1996 2012-04-02T13:05:01Z 2012-04-02T16:58:46Z <p>When working with group cosets in MAGMA is there a way of treating the cosets as subsets of the overlying group. Specifically I have a group $G$ and subgroups $H$ and $K$ . I wish to look at the intersection of a pair of cosets $Hh$ and $Kk$ for some $h,k\in G$ , but am unable to perform such operations in MAGMA when they are considered as cosets.</p> http://mathoverflow.net/questions/92891/working-with-group-cosets-in-magma/92909#92909 Answer by Dima Pasechnik for Working with group cosets in MAGMA Dima Pasechnik 2012-04-02T16:33:39Z 2012-04-02T16:33:39Z <p>Well, this is trivial in <a href="http://www.gap-system.org/" rel="nofollow">GAP</a>. Here is an example:</p> <pre><code>gap&gt; G:=SymmetricGroup(7);; gap&gt; H:=Stabilizer(G,1);; gap&gt; K:=SylowSubgroup(G,2);; gap&gt; c1:=RightCoset(H,(1,2));; gap&gt; c2:=RightCoset(K,(1,2,3));; gap&gt; Intersection(c1,c2); [ (1,2,3), (1,2,3)(5,6), (1,2,3,4), (1,2,3,4)(5,6) ] gap&gt; </code></pre> <p>By the way, GAP is free, unlike Magma...</p> http://mathoverflow.net/questions/92891/working-with-group-cosets-in-magma/92910#92910 Answer by Derek Holt for Working with group cosets in MAGMA Derek Holt 2012-04-02T16:58:46Z 2012-04-02T16:58:46Z <p>As far as I can see, the only way to do that directly with cosets $C1$ and $C2$ of $G$ is</p> <p><code>$\{ x : x\ {\rm in}\ G\ |\ x\ {\rm in}\ C1\ {\rm and}\ x\ {\rm in}\ C2 \}$</code></p> <p>which looks very inefficient, because it is iterating over all of $G$.</p> <p>I would suggest first find a right transversal $T$ of $H \cap K$ in $H$, and then search through $T$ looking for an element $t \in T$ with $thk^{-1} \in K$. If you find such a $t$, then the intersection is the coset $(H \cap K)th$, and otherwise it is empty.</p>