$F_2$ 包含一个与 $F_n$ 同构的子群

hbghlyj

$F_n=⟨x_1,\dots,x_n⟩$显然包含一个与$F_2$同构的子群$⟨x_1,x_2⟩$.反过来也成立$F_2$包含一个与$F_n$同构的子群

UTM groups and symmetry, Armstrong

27.10. Let $n$ be a positive integer. Prove that $F_2$ contains a subgroup which is isomorphic to $F_n$.

只要用前一题(27.9)便解决了

27.9. If $x, y$ generate $F_2$, let $$ X=\left\{x y x^{-1}, x^2 y x^{-2}, x^3 y x^{-3} \ldots\right\} $$ and let $H$ denote the subgroup generated by $X$. Show that $X$ is a free set of generators for $H$.

如何证27.9 我的想法是将 $X$ 的两个不同元素相乘时$(x^i y x^{-i})(x^j y x^{-j}),i\ne j$中间的$x^{j-i}\ne e$不会消去

hbghlyj

@DonAntonio 的回答

Hints:

1) Prove that $\,F_\infty=$ the free group on a countable set, contains $\,F_k=$the free group on $\,k\;,\;\;k\in\Bbb N\,$ generators

2) Prove that $\,F_2^{'}:=[F_2:F_2]\cong F_\infty\,$ , with $\,F_2:=\langle x\,,\,y\;;\;\emptyset\rangle\,$ , by showing that

$$F_2^{'}=\langle \,[x^n\,,\,y^m]\;;\;n,m\in\Bbb Z-\{0\}\,\rangle$$

Note: You may try to prove that $\,F_2/F_2^{'}\cong \Bbb Z^\infty\,$

Hint for a hint: Theorem 2.10 in the classical "Combinatorial Group Theory...", by Magnus, Karrass & Solitar gives you another very nice way to prove (2) above. This book is a must in every group theory lover's library

Show that for each integer $n ≥ 2$, $F_n$ is a finite index subgroup of $F_2$, where $F_n$ is the free group on $n$ generators.

Czhang271828

对每个 $F_n\subseteq F_2$ 中的有限的字 $xyxy\cdots xy$, 从左至右地将 $x^ly$ 写作 $x^lyx^{-l}\cdot x^l$ 即可. 例如
\[
xy^2xyx^{-2}\longrightarrow xyx^{-1}\cdot xyx^{-1}\cdot x^2yx^{-2}.
\]
显然这种写法是唯一的, 并且可以通过 $x^kyx^{-k}\mapsto g_k$ 对应至自由群 $F_n$.

hbghlyj

Czhang271828 发表于 2023-5-6 07:40
对每个 $F_n\subseteq F_2$ 中的有限的字 $xyxy\cdots xy$

缺少指数觉得应该是           $\smash{\raise24.5em{\bbox[#fff,4pt]{x^{\alpha_1}y^{\alpha_2}\dots}}}$

hbghlyj

见Theorem 2. For $m$ and $n$ non-negative integers, does $F_m$ have a subgroup isomorphic to $F_n$?

Proof. At first suppose $n=1$, then $F_1 \cong \mathbb{Z}$ is abelian. So we can let $m=1$ and then we have $F_1 \cong F_1$. Notice that if $m>1$, $F_m$ is not an abelian group, so it cannot have a subgroup isomorphic to $F_1$.

Now let $n \geqslant 2$. Notice that for the map $F_1 \rightarrow F_n$ sends the generating element of $F_1$ to the generating element of $F_n$. This is an injective homomorphism, so $F_1$ has a subgroup isomorphic to $F_n$. Similarly for the map $F_2 \rightarrow F_n$ sends the two elements of the generating set of $F_2$ to the generating set of $F_n$. Notice that this is also an injective homomorphism, so $F_2$ has a subgroup isomorphic to $F_n$.

蓝色句是不是说反了

hbghlyj

Czhang271828 发表于 2023-5-6 07:40
对每个 $F_n\subseteq F_2$

在这种构造怎样证明$|F_2:F_n|$有限呢