Maths - Free Product

Free Product (Coproduct, or Categorical Sum)

Each multiplicand is represented by a 'word' consisting of a sequence of generators. We apply the multiplication by putting the words one after the other. This is the most general type of product that contains the two multiplicands as subgroups.

We can 'reduce' the result by:

• removing the identity element
• replacing and sequence that is within one of the multiplicands

but unless we apply any specific reductions the result will always be infinite for non trivial multiplicands.

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
<m | m²>
 1 m m 1

C3

generator cayley graph table
<r | r³>
 1 r r² r r² 1 r² 1 r

Free Product C3 × C2

So what would be the elements of this group? Lets assume they are the same as for the direct product, that is: 1,r,r²,m,rm and r²m. Using these as words would give the table:

 1 r r² m rm r²m r rr rr² rm rrm rr²m r² r²r r²r² r²m r²rm r²r²m m mr mr² mm mrm mr²m rm rmr rmr² rmm rmrm rmr²m r²m r²mr r²mr² r²mm r²mrm r²mr²m

We can then apply any constrains for the individual groups, that is: m²=1 and r³=1 which gives:

 1 r r² m rm r²m r r² 1 rm r²m m r² 1 r r²m m rm m mr mr² 1 mrm mr²m rm rmr rmr² r rmrm rmr²m r²m r²mr r²mr² r² r²mrm r²mr²m

But we still get elements that are not members of the group (the group is not closed). If we make these elements of the group, then we have to make them rows and columns, in which case more elements get added to the body of the table. So the group is infinite. The elements of the group are alternating r's and m's such as:

r,m,r,m,r,m,r,m…

or

m,r,m,r,m,r,m,r…

where r may be replaced by r² at any place.

generator cayley graph table
<m,r | m²,r³>

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.