Does anyone know what this notation means? Group Theory/Set Theory "<>" My professor is away and I get an automated reply that she can't answer :/
I am asked to show a particular group G is isomorphic to $H=\langle x,y \mid x^3,y^4,yx=x^2y\rangle$ which I don't understand.
First, I understand the notation $\langle x,y\rangle$ to be (assuming $x,y$ are as they are, i.e. a monomial) a "span" of them i.e. $\langle x,y\rangle =\{c_1x+c_2y\mid c_1,c_2 \in k[x,y]\}$ where $k$ is a field.
But this is a group theory question so I'm a little confused as to what the brackets mean.(No, I cannot find similar notations in my lecture notes...)
Also, What does $x^3,y^4$ mean? I mean, are they equal to the identity(whatever that is)? There's no "=" sign here I don't know what $x^3$ or $y^4$ on its own would possibly mean. I am taking "|" as "such that".
I am wondering if any experienced mathematicians have an idea what this group is supposed to be. Any ideas??
 A: No, in this context $\langle x,y\rangle$ means the free group with the two generators $x$ and $y$, and $\langle x,y\mid x^3,y^4, yx=x^2y\rangle$ is the quotient of that free group by the normal subgroup generated by the elements $x^3$, $y^4$, and $yx(x^2y)^{-1}$.
Your group theory notes ought to have a section explaining "generators and relations".
A: This means "a group with generators $x$ and $y$, and relations $x^3 = e, y^4 = e, yx = x^2 y$."
Roughly: take any word with only $x$s and $y$s, like $xxyxyyx$, which we write as $x^2yxy^2x$ (and also allow negative exponents, none of which show up in this example). You're allowed to simplify that with the rules. For instance, 
$x^3 = e$ means that $x^2 = x^{-1}$, so we can say 
$$
x^2yxy^2x = 
x^{-1}yxy^2x
$$
Alternatively, we can replace $x^2y $ with $yx$, to get
$$
x^2yxy^2x = 
yxyxy^2x
$$
Through similar operations, you may be able to convince yourself that there are only finitely many distinct elements in the group. 
More formally, though, you look at the free group $F$ on the letters $x,y$ (i.e., all finite strings in $x, y, x^{-1}, y^{-1}$, subject to only the obvious relations like $xx^{-1} = e$ (the empty string). 
This has a subgroup $R$ generated by the elements $xxx, yyyy, yxy^{-1}x^{-2}$. The quotient of $F$ by $R$ is the group $G$. 
There's a really nice presentation of this material in Massey's "Algebraic Topology: An Introduction", in which he discusses the history -- in which saying things like what I started with was pretty common, until people figured out it didn't really make sense, and formalized it, and then rewrite it all in terms of universal properties, etc, which make for easy proofs but often not great understanding, at least not if you haven't seen it done the basis way first. :)
