Wikipedia:Reference desk/Archives/Mathematics/2015 October 15

= October 15 =

free group question
Hello. Suppose I have a free group with $$X=abcde$$ and $$Y=klmnopq$$, say. Is there a word or concept for $$abYcde$$? I want to do some of X, then Y, then the rest of X. I happen to know that $$X^2=Y^2=id$$, if that's important. thanks in advance, Robinh (talk) 09:09, 15 October 2015 (UTC)
 * First, I'm pretty sure you can't have an element of order 2 in a free group. But leaving that aside, I don't think there is a name for what you have in mind, though you can have a concept for anything you can define. Maybe you could define it in symbols as Z=PYP-1X. I'm not sure that the phrase "some of X, ..., rest of X" makes sense since you might be able to take X to be one of the generators. For example if G is the free group on {a, b, c}, X=ab, then G is also free on {X, b, c}, so your Z could have two or three possible values depending on which set of generators you use. --RDBury (talk) 10:01, 15 October 2015 (UTC)
 * (OP) thanks for this. You are right that my group is not free (there are loads of relations but the only possibly relevant ones are $$X^2=Y^2=id$$).  The generators are the lower-case letters $$a,b,c,\ldots$$.  Part of my problem is trying to express the idea, as you identify.  I use the notations $$ x^y=y^{-1}xy$$ and $$\left[x,y\right]=x^{-1}y^{-1}xy$$ which I find very helpful, and I wondered if there was something comparable for $$abYcde$$. Best wishes, Robinh (talk) 18:59, 15 October 2015 (UTC)
 * Actually I don't think group theory has many notations for operations with single elements. There is xn, xy and the commutator [x, y] you already mentioned, plus iterated commutators are seen sometimes. There are plenty of notations involving sets or defining subgroups, like G', Cl(g), NG(S), and others, so apparently it's more useful to think in terms of them. --RDBury (talk) 06:11, 16 October 2015 (UTC)
 * OK, so it looks as though we need some notation. How about $$abYcde={}^X_2Y$$ where the 2 means take the first two elements of X.   Anyone got any better ideas for a nice notation? Robinh (talk) 08:19, 16 October 2015 (UTC)