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

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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, x^y 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), N_G(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 $abYcde={}_{2}^{X}Y$ 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)