Maths - Bicrossed Product

Direct Product

Perhaps the simplest way to combine (multiply) two sets (and therefore groups) is to use the Cartesian product as explained on this page. This product produces a set from the product:

g×h = {g, h}

So the result of this product is a different type of entity than the elements being multiplied, so the multiplication is not closed, and therefore does not represent a group. The external product makes this into a group because the inputs to the multiplication are also sets:

{y,z}{y',z' } = {y(z • y'),z^y' z'}

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 *.

Example C₂×C₃

In order to try to understand this product of two groups lets try multiplying two very simple groups together, the simplest groups I can think of are C₂ and C₃.

C₂

generator	cayley graph	table	permutation	representation
<m | m²>		1 m m 1	< ( 1 2 ) >	0 1 1 0

C₃

generator	cayley graph	table	permutation	representation
<r | r³>		1 r r² r r² 1 r² 1 r	< ( 1 2 3 ) >	0 0 1 1 0 0 0 1 0

direct product C₃ × C₂

This gives :

generator	cayley graph	table
<m,r | m²,r³,rm=mr>		{1,1} {r,1} {r²,1} {1,m} {r,m} {r²,m} {r,1} {r²,1} {1,1} {r,m} {r²,m} {1,m} {r²,1} {1,1} {r,1} {r²,m} {1,m} {r,m} {1,m} {r,m} {r²,m} {1,1} {r,1} {r²,1} {r,m} {r²,m} {1,m} {r,1} {r²,1} {1,1} {r²,m} {1,m} {r,m} {r²,1} {1,1} {r,1}
permutation	representation
<(1 2 3)(4 5 6),(1 4)(2 5)(3 6)>	[ 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 , 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ]

Note that, in addition to applying both the generators and constrains for the original groups we have had to apply an additional constraint: rm=mr. If we had not done this we would have the infinite free product.

Generating a Bicrossed Product using a Program

We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.

EuclideanSpace

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