Maths - External Group Product

here the group elements of the resulting group are sets containing an element from each on the multiplicands.

Example 1

Lets take the simplest example we can think of. Lets take the group Z2 which is the integers modulo two which gives an exclusive or table:

Cayley Table
Cayley Graph
  0 1
0 0 1
1 1 0
cayley digraph z2

Lets combine two of these groups to give: Z2⊕Z2

{g, h} × {g' , h' } = {g * g' , h o h' }

where:

Cayley Table
  {0,0} {0,1} {1,0} {1,1}
{0,0} {0*0,0o0}={0,0} {0*0,0o1}={0,1} {0*1,0o0}={1,0} {0*1,0o1}={1,1}
{0,1} {0*0,1o0}={0,1} {0*0,1o1}={0,0} {0*1,1o0}={1,1} {0*1,1o1}={1,0}
{1,0} {1*0,0o0}={1,0} {1*0,0o1}={1,1} {1*1,0o0}={0,0} {1*1,0o1}={0,1}
{1,1} {1*0,1o0}={1,1} {1*0,1o1}={1,0} {1*1,1o0}={0,1} {1*1,1o1}={0,0}
Cayley Graph cayley digraph z2+z2

Is there any way we can modify this to generate the complex numbers?

 

Can we then go on to generate the quaternions and octonions?


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Correspondence about this page

Book Shop - Further reading.

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flag flag flag flag flag flag Contemporary Abstract Algebra written by Joseph Gallian - This investigates algebra (Groups, Rings, Fields) using an abstract (set theory) approach. It is very readable for such an abstract topic but the inclusion of 1 page histories of the great mathematicians and other background information.

 

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