Are all the words in the free group, or just the freely-reduced words?
If the latter, does saying that the group operation involves free reduction imply that only freely-reduced words will end up in the free group? Because, by default I would assume that "the group has as its element the words over X∪X−1" means that all the words (including non-freely-reduced ones) are in the free group.
It sounds like you didn't already know what the free group is; in that case (and even if you did already know), it's very gratifying to know that someone is actually reading this carefully!
You're quite right to flag this up; I was being sloppy. There are three main ways to construct the free group, and I've kind of mixed together the two of them which are most intuitive. I'm trying not to simply define the free group here, but you're right that I've done it confusingly. I'll fix it.
Searching Google Books (and Wikipedia), I don't see precedents for use of "freely reduced" rather than "reduced".
Now, inventing and using good terminology is a good thing. What do you think about expanding on the choice of term in the article text?
Are all the words in the free group, or just the freely-reduced words?
If the latter, does saying that the group operation involves free reduction imply that only freely-reduced words will end up in the free group? Because, by default I would assume that "the group has as its element the words over X∪X−1" means that all the words (including non-freely-reduced ones) are in the free group.
Probably more like Math 2 or 2.5. I'm a programmer without any formal training in math beyond what's required to get an electrical engineering degree.
Occasionally I skim math blogs out of curiosity, but frequently don't understand what I'm reading.
Are you otherwise broadly Math 3? It would be good to have a guinea pig for group theory.
That's right. I know very little group theory.
I was just remarking to Stephanie how I was able to understand everything on this page, right before I got to that sentence that I found confusing :-)
It sounds like you didn't already know what the free group is; in that case (and even if you did already know), it's very gratifying to know that someone is actually reading this carefully!
You're quite right to flag this up; I was being sloppy. There are three main ways to construct the free group, and I've kind of mixed together the two of them which are most intuitive. I'm trying not to simply define the free group here, but you're right that I've done it confusingly. I'll fix it.