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

Lets combine two of these groups to give: Z₂Z₂

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

where:

• × is the operation of the combined algebra.
• * is the operation of the group G.
• o is the operation of the group H which may be, or may not be, the same as *.

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

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

 

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

• H = CC
• O = HH

EuclideanSpace

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Generating Geometric Algebras