1) Do generators exist for S3 (the symmetric group)? I thought the symmetric groups all had generators?
2) When working out products of elements in S3 I get that (231)(213) = (123). This is a product of two 3-cycles and they are not inverses so (123) has to represent a 3-cycle. How do I tell otherwise that this product isn't the identity?
3) When working out the multiplication table I noted several elements that are equivalent, such as (132) and (213). But I can't make this identification lest the table have multiple instances of (132) in a single row of the table. However this implies that the elements of S3 must contain (132) and (213), which makes no sense if |S3| is only 6.
Gaaaaaaaaaaaaaaaaah!!!! This was supposed to be an easy exercise!
Okay, I'm seeing things a bit better now.
Thanks for the help, johng, it's a lot clearer. Though if I may inquire, does S3 have a set of generators? I still haven't been able to find any.
Any group G is said to be finitely generated iff there is a finite set such that every element of G is a "word" in the generators . Here a word is a product of k elements of G, where each is one of the and is an integer (positive, negative or 0); k need not be the same for different group elements.
Trivially, then any finite group is finitely generated -- just take the generating set to be the entire group. An infinite direct sum is not finitely generated.
However, I think what you want is a non=trivial generating set for . Let a=(12) and b=(123). Then e=a0, (12)=a1, (13)=(12)(123)=ab, (23)=(12)(123)2=ab2, (123)=b and (132)=(132)(132)=b2.
is said to have presentation by generators and relations: . As above . The group table is then computable from the relations. For example, ; the 2nd equality since a2=1. Oops, I automatically used symbol 1 for the identity; 1 and e are synonyms as group elements.
Typically, this is the way finite groups are described; i.e. via generators and relations. You need the concept of free groups to see that any finite group has a presentation.
One last example: Let n be a positive integer greater than or equal to 2. The dihedral group . is the group of symmetries of a regular n-gon. As you can see, .
You might try and convince yourself that and then work out the product of any two elements.