Thread: Smooth? View Single Post
2004-10-31, 08:03   #2
Chris Card

Aug 2004

2×5×13 Posts

Quote:
 Originally Posted by Xyzzy In several threads I have come across a term that I do not understand: "smooth group order" What does this mean? Not only do I not know what "smooth" means, I have no idea what a "group order" is... Thanks!
An integer is said to be "smooth" if all its prime factors are smaller than a given bound (the bound usually being clear from the context).

A group is an abstract mathematical object consisting of a set and an operation that produces a member of the set from pairs of members of the set, and which satisfies 4 axioms (I won't give all the details here). The set could be finite or infinite, but in the case that it is finite, then the number of elements in the set is called the "order" of the group, or the "group order". A simple example of a group is the set of integers {1, ..., p-1}, p a prime number, with the operation being multiplication modulo p, and in this case the group order is p-1.

So, "smooth group order" refers to a finite group whose order is smooth. One place this comes up is in the P-1 factorisation method, where the success of the method depends on the group in the example described above having smooth group order.

HTH

Chris