Vierergruppe

Author

Eric W. Weisstein
December 5, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/Vierergruppe.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Vierergruppe.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Permutation Group

pg=Join[{{1,2,3,4}},FromCycles/@{{{1,2},{3,4}},{{1,3},{2,4}},{{1,4},{2,3}}}]

pgrules=Thread[pg->{"I","(Subscript[V, 1])","(Subscript[V, 2])","(Subscript[V, 3])"}]

nrules=Thread[pg->Range[4]]

Multiplication Table

GroupMultiplicationTable[pg,pgrules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[pg,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[V],
DisplayFunction->Identity]&/@{Hue[#/2]&,GrayLevel}]]

Subgroups

Subgroups[pg]/.pgrules

NormalSubgroups[pg]/.pgrules

Classes

ConjugacyClasses[pg]/.pgrules

Properties

Through[{SimpleGroupQ,AbelianGroupQ}[pg]]

GroupDataString

GroupDataString[pg,{"Vierergruppe","Vierergruppe"},{},60]