Questions tagged [free-product]

In mathematics, specifically group theory, the free product is an operation that takes two groups $G$ and $H$ and constructs a new group $G\ast H$. The result contains both $G$ and $H$ as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties

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A certain free product of groups is VTF

Suppose that $G_1,\ldots,G_n$ are finite groups, and $m\geqslant 0$ is some integer. Set $$G=G_1\ast\cdots\ast G_n*F_m,$$ (where $F_m$ is the free group on $m$ generators). Then, is $G$ virtually ...
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If $A\ast_C B$ is finitely generated, are $A$ and $B$ finitely generated?

I know that if $A$ and $B$ are group of finite rank $n$, there is an amalgamated product $A\ast_{F_2} B$ of rank $2$. It is know that for every $n$ there exist groups $A$ and $B$ of rank $\ge n$ and ...
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Presentation of the amalgamated product (Van Kampen's theorem)

Supposing $\pi_1(U_1)=\langle S \mid R\rangle$ and $\pi_1(U_2)=\langle U \mid V\rangle$, would: $$\pi_1(U_1)*_{\pi_1(U_1\cap U_2)}\pi_1(U_2)=\langle S,U \mid R,V, T\rangle$$ where $T$ is the ...
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Fundamental group of that using Seifert-van Kampen

I have this exercise: Compute the fundamental group $\pi_1(X)$ of the space $X=S^2 \cup \{ (x,0,0) : x\in [-1,1] \} \cup \{(0,y,0):y\in [-1,1] \} \cup \{(0,0,z):z\in [1,1] \}$. I tried that via ...
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Free product not abelian

If $(G_i)$, $i \in I$ with $|I| > 1$ and $|G_i| > 1$ is the free product $G$ of $(G_i)$,$i \in I$ always not-commutative? The free product of a family of groups $(G_i)$, $i \in I$ is defined by ...
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Finding all the subgroups of $\mathbb{Z}_2*\mathbb{Z}_2$

Consider the group that is the free product of two copies of $\mathbb{Z}$, that is $G=\mathbb{Z}_2 * \mathbb{Z}_2=\left\langle a,b\;\vert \;a^2=b^2=e \right\rangle$. I'm trying to find all of its ...
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If $N$ is the least normal subgroup of $A*B$ containing $A$, then $(A*B)/N \cong B$.

If $N$ is the least normal subgroup of $A*B$ containing $A$, then $(A*B)/N \cong B$. My Proof: Let $f:A \to B$ be the homomorphism given by $f(a) = e_B$. Note that $1_B:B \to B$ is also a ...
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Can a Free product with amalgamation of $\mathbb{Z}*\mathbb{Z}$ be isomorphic to $\mathbb{Z}\times \mathbb{Z}$?

I was trying to compute the fundamental group of torus by forcing myself of using the Seifert Van Kampen theorem. One knows that the answer is the direct product of $\mathbb{Z}$ and $\mathbb{Z}$. ...
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Determining whether an element of a free product of cyclic groups is a commutator.

Let $G=C_{n_1}*\cdots*C_{n_k}=\langle a_1,\cdots,a_k\mid a_1^{n_1}=\cdots=a_k^{n_k}=1\rangle$ be a free product of finitely many finite cyclic groups. Given a word $g=g_1\dots g_n\in G$, is there an ...
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Are "free products" just coproducts in categories admitting presentations of objects (generators and relators)?

For groups, the "free product" can be taken "generator-wise" and "relator-wise" as done here: https://ncatlab.org/nlab/show/free+product+of+groups It is also the case that the "free product" is the ...
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is the preimage of a free product of groups itself a free product

I have an epimorphism of groups $f:G\to H*K$. Is it true that $G=H'*K'$ for the preimages $H'=f^{-1}(H)$ and $K'=f^{-1}(K)$? UPDATE: (after Tsemo's answer) I have a group $G$ generated by two of its ...
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If $G = H$ is a non-trivial group, is it possible for $G*H$ to be isomorphic to $\mathbb Z^2$ [closed]

If $G = H$ is a non-trivial group, is it possible for $G*H$ to be isomorphic to $\mathbb Z^2$
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Wedge sum of circles

Is the wedge sum of circles considered as a path? According to Wikipedia(Rose), the figure eight graph $S^1\vee S^1$ is directed even though it is the direct sum of two circles with two points ...
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On the converse of extension lemma of external free product --- Munkres Lemma 68.5

This is Lemma 68.5 in Munkres, which states as follows Let $\{G_{\alpha}\}_{\alpha\in J}$ be a family of groups; let $G$ be a group; let $i_{\alpha}:G_{\alpha}\longrightarrow G$ be a family of ...
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Is random homomorphism from free product of $k$ copies of $\mathbb Z_2$ to orthogonal group $O(3)$ injective?

Consider the group $G = \ast_{i=1}^k \mathbb Z_2 = \langle (g_i)_{i=1}^k : g_i^2 = 1 \rangle$. Suppose we define a homomorphism $\phi$ from $G$ to $O(3)$ by setting $\phi(g_i)$ to the reflection in a ...
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What is the free product in the category of operads?

Is the Hadamard product (5.3.3 in Loday-Vallette's book on Algebraic operads) the free product in the category of operads? Is there a free product defined in the category of 1/2props or props?
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Dehn twist in amalgamated product, preserving a subgroup

I'm in an uncertain position where I have the following: $G=A \underset{C}{\ast} B$ and $H\leq G$. Let $h\in H \cap C$ where $C$ is abelian. Define, \tau\left(g\right)=\begin{cases} g & g\in ...
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How to prove that the following groups are isomorphic?

Consider the following groups. $\langle a,b,c~ |~aba^{-1}b^{-1},aca^{-1}c^{-1}\rangle$ $(\Bbb Z * \Bbb Z) \times \Bbb Z$ $((\Bbb Z \times \Bbb Z)*(\Bbb Z \times \Bbb Z))/N$, where $N$ is the normal ...
In fact, the free product of two finitely presented Lie algebras is also a finitely presented Lie algebra. Let consider the definition of dialgebras: A diassociative algebra is a $K$-linear space, ...