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I'm trying to learn abstract algebra, and I'm stuck on these two problem, both about free groups - specifically, both about the free group of 2 generators.

So let the group be $\mathbb{F}_2=\langle x,y \rangle$. The questions are:

First problem

The first question is to show is that the subgroup $G=\langle x^2, xy\rangle$ is also free. I have no idea how to approach it.

Second problem

The second problem is to find a subgroup of $\mathbb{F}_2$ which is isomorph to $\mathbb{F}_\omega = \langle u_i \rangle_{i\in\mathbb{N}}$

My approach has been this - I want to find a formula for the generator $u_i$ in terms of $x,y$ - so basically to "encode" the group with $x,y$. However, everytime I try, either exponentiating a generator doesn't do what it should, or multiplying two distinct generators doesn't do what it should, or inversion doesn't work, etc.

I've asked this in another math-help forum, and despite getting some pointers, nothing really helped me.

Thanks in advance.

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    1. Subgroups of free groups are free, but if you don't know that, any relatipn between $x^2$ and $xy$ would also be be trivial in $F(x,y)$. That means, at the least, the exponents of $x$ and $y$ would be zero. What does that tell you about your relation? – 2017-01-19
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    2. Consider the group generated y $x^iyx^{-i}$ for all positive integers $i$. Can you show this is free? – 2017-01-19

1 Answers 1

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Hints:

First problem: Let $a=x^2$ and $b=xy$. Then $G\cong F(a,b)/\langle\langle R\rangle\rangle$ (where $F(a,b)\cong\mathbb{F}_2$ is the free group on $a$ and $b$). So an element in $G$ looks like $x=x_0^{\pm 1}x_1^{\pm 1}\cdots x_n^{\pm 1}$ where $x_i\in\{a,b\}$ for each $i$. Show by induction on $n$ that $x=1$ in $G$ if and only if $x=1$ in $F(a,b)$.

Second problem: Consider conjugation of $x$ by $y$ (that is $y^{-1}xy$).