Maths - 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:

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

where:

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. As a reminder these are the definitions of these groups individually (full definitions on this page):

C2

generator cayley graph table permutation representation
<m | m²> c2 graph
1 m
m 1
< ( 1 2 ) >
0 1
1 0

C3

generator cayley graph table permutation representation
<r | r³> c3 graph
1 r
r 1
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> c3c2 graph
{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 Direct Product using a Program

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

(1) -> )r axiom/directProduct
)set output algebra off
)set output mathml on
-- first calculate C2 x C3
C2 := FiniteGroup(2,[[1,2],[2,1]],["1","m"])
                                                               Type: Type
DP := directProduct([[1,2,3],[2,3,1],[3,1,2]],["1","r","rr"])$C2

                                                               Type: Type
toTable()$DP

1 r rr m mr mrr
r rr 1 mr mrr m
rr 1 r mrr m mr
m mr mrr 1 r rr
mr mrr m r rr 1
mrr m mr rr 1 r
                      
                           

                                                           Type: Table(6)
setGenerators([false,true,false,true,false,false])$DP

                                                               Type: Void
PDP := toPermutation()$DP

<(1 2 3)(4 5 6),(1 4)(2 5)(3 6)>
                           
                                

                                          Type: PermutationGroup(Integer)
permutationRepresentation(PDP,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
]
                      
                           

                                              Type: List(Matrix(Integer))
-- now calculate C3 x C2
C3 := FiniteGroup(3,[[1,2,3],[2,3,1],[3,1,2]],["1","r","rr"])

                                                               Type: Type
DP := directProduct([[1,2],[2,1]],["1","m"])$C3

                                                               Type: Type
toTable()$DP

1 m r rm rr rrm
m 1 rm r rrm rr
r rm rr rrm 1 m
rm r rrm rr m 1
rr rrm 1 m r rm
rrm rr m 1 rm r
                      
                           

                                                           Type: Table(6)
setGenerators([false,true,true,false,false,false])$DP

                                                               Type: Void
PDP := toPermutation()$DP

<(1 2)(3 4)(5 6),(1 3 5)(2 4 6)>
                           
                                

                                          Type: PermutationGroup(Integer)
permutationRepresentation(PDP,6)

[
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 1 0 0
0 0 1 0 0 0
0 0 0 0 0 1
0 0 0 0 1 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
0 0 0 1 0 0
]


                                              Type: List(Matrix(Integer))
(13) ->

 


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