example of groups of order pq

As a specific example, let us classify groups of order 21. Let G be a group of order 21. There is only one Sylow 7-subgroup K so it is normal. The possibility of there being conjugate Sylow 3-subgroups is not ruled out. Let x denote a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for K, and y a generator for one of the Sylow 3-subgroups H. Then x7=y3=1, and yxy-1=xi for some i<7 since K is normal. Now x=y3xy-3=y2xiy-2=yxi2y-1=xi3, or i3=1mod7. This implies i=1,2, or 4.

Case 1: yxy-1=x, so G is abelianMathworldPlanetmath and isomorphicPlanetmathPlanetmathPlanetmath to C21=C3×C7.

Case 2: yxy-1=x2, then every product of the elements x,y can be reduced to one in the form xiyj, 0i<7, 0j<3. These 21 elements are clearly distinct, so G=x,yx7=y3=1,yx=x2y.

Case 3: yxy-1=x4, then since y2 is also a generator of H and y2xy-2=yx4y-1=x16=x2, we have recovered case 2 above.

Title example of groups of order pq
Canonical name ExampleOfGroupsOfOrderPq
Date of creation 2013-03-22 14:51:15
Last modified on 2013-03-22 14:51:15
Owner jh (7326)
Last modified by jh (7326)
Numerical id 4
Author jh (7326)
Entry type Example
Classification msc 20D20