# Questions tagged [free-groups]

Should be used with the (group-theory) tag. Free groups are the free objects in the category of groups and can be classified up to isomorphism by their rank. Thus, we can talk about *the* free group of rank $n$, denoted $F_n$.

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### Prove that $\text{Out}(F_2) \cong \text{GL}_2(\mathbb{Z})$

Let $F_2$ be a free group of rank $2$. Define: $\text{Out}(F_2)=\text{Aut}(F_2) / \text{Inn}(F_2)$. Prove that: $$\text{Out}(F_2) \cong \text{GL}_2(\mathbb{Z})$$ In previous exercise, I showed that: ...
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### Meaning of cut vertex in Whitehead graph in the sense of language

I recently reading Whitehead algorithm in paper of Stallings (without bothering 3-manifolds). Whitehead graph and the cut vertex play an important role. Let us consider one example from Stallings ...
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### Can we construct a free structure on a non associative algebraic structure.

For any set we can construct a free group on it. Also for non associative structures like Lie algebra, Lie ring we may construct free structures, but these are non associative structures and having ...
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### Virtual solvability of dense subgroups

Let $G$ be a (finitely generated) dense subgroup of $\mathsf{SL}(2;\mathbb{C})$. Is it possible that $G$ is virtually solvable? In other words, by Tit's alternative, does being dense necessitate the ...
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### Exercise on Generators and Relations from Michael Artin's book

The question is: Let $\phi: G \mapsto G'$ be a surjective group homomorphism. Let $S$ be a subset of $G$ whose image under $\phi$(S) generates $G$', and let $T$ be a set of generators of $\ker\phi$. ...
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### Burnside groups with GAP system [closed]

My question is related to Burnside groups $B(n, 3)$ in the GAP system. I'm interested in ways to represent Burnside groups $B(n, 3)$ in GAP. The obvious representation using relations (see example for ...
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### Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same?

I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student ...
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### Are right-adjoints of a forgetful functor reflectors?

From what I understand, there is no formal definition of a forgetful and an inclusion functor, but more like "moral guidelines" with "good properties" of why we would call them ...
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### Bipartite intersection graph

Recently, I'm trying to learn geodesic currents on free groups through a paper by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form ...
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