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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Amalgamated free product and van Kampen

Observe the two homomorphisms $\varphi_1 : A\to G$ and $\varphi_2 : A \to H$ of a group $A$ into the groups $G$ and $h$. The set $N=\{\varphi_1(a)\varphi_2(a)^{-1} : a\in A\}$ is a normal subgroup of $...
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Why is the free product of a group with itself not isomorphic to itself?

I've been recently introduced to the definition of free product of groups, and I'm struggling to identify the details of its precise definition. In particular, it seems to me from the definition I've ...
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Using Kurosh’s Theorem to study elements of $PSL(2, \mathbb{Z})$

Ok, I believe this is a relatively basic question, but I couldn’t figure out what I’m doing wrong. What I’d like to prove is the following result: If an element $T$ in $SL(2, \mathbb{Z})$ has finite ...
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Relations from quotient of free product

This question arose from an exercise which asks you show that the fiber coproduct exists in the category of groups. I was eventually able to (mostly) solve the problem by “gluing” the images of ...
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Is the coproduct of any family of groups a free group

In Lang’s Algebra, he constructs the coproduct of a family of groups in a similar way to the construction of the free group of a set $S$, $F(S)$. He later constructs the free product of $n$ groups and ...
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Amalgamated Product, $G_1*_HG_2$ is surjective $\iff$ $G = \langle G_1\coprod G_2\rangle$

Suppose we are given groups $G_1,G_2$ and some subgroup $H$ of $G_1$ and $G_2$ and let $G_1*_HG_2$ denote the amalgamated free product. I stumbled upon the following claim: $G$ is generated by $G_1\...
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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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In van Kampen’s theorem, what happens to the loops not in $\pi_1(U_\alpha \cap U_\beta)$?

I’ve just read the set up and the statement for van Kampen’s theorem. Here’s the version that we use in our class. van Kampen's Theorem. Suppose we have an open cover $\left\{U_{\alpha}: \alpha \in A\...
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Good introduction to free groups and free products

In my undergraduate research project, I am going to study a paper on free products in division rings. To do this, however, I, of course, need to learn about free groups and free products. Right now, ...
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Relation between finite residuals and free products.

Question: Let $G$ be a group. The finite residual of $G$ is the intersection of all finite index subgroups of $G$. We denote this with $\text{fr}(G)$. Note that $\text{fr}(-)$ is functorial on the ...
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Free Products - representation of 1 by empty word is unique

I am studying Free Products of Groups from Munkres' Topology book. The definition of free products is: Let $G$ group, $\{G_a\}_{a \in J}$ family of groups that generates $G$ and $G_a \cap G_b$ ...
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Characterization of free product of groups by lifting property with respect to morphisms with right inverse

A morphism $i : A \to B$ has the left lifting properties with respect to a morphism $p : X \to Y$ if and only if any commutative square with $i$ on the left and $p$ on the right has a diagonal lift ...
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Are the central quotients of braid groups non-trivial free products?

The braid group $B_3$ has the property that its central quotient (i.e., $B_3 / Z(B_3)$) is isomorphic to the modular group $\mathrm{PSL}(2,\mathbb{Z})$. The modular group is known to be isomorphic to ...
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Presentation of the amalgamated product of $G_1$ and $G_2$ above $H$ is $ \langle S_1,S_2\; ; \; R_1,R_2,\phi_1(s)\phi_{2}^{-1}(s),s \in S \rangle $

Here, I found the proof of presentation of the free product of groups. I wanted to show the same thing for amalgamated free product of 2 groups i.e. Show that that if $H$ is generated by $S$, and $ \...
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Subgroup of $\mathbb{Z}\ast \mathbb{Z}$

I want to see $\mathbb{Z}\ast\mathbb{Z}$ has a subgroup of index 4 which is not normal and I want to calculate the rank of this subgroup. One way is to classify all the subgroups of $\mathbb{Z}\ast\...
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1answer
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Free groups as free product of infinite cyclic groups

Let $S$ be an arbitrary set (countable or uncountable). It is clear that the free abelian group generated by $S$ is isomorphic to the direct sum $$\bigoplus_{s\in S}\mathbb{Z}.$$ Is the free group ...
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Homomorphisms between $\mathbb{Z} \star \mathbb{Z} $ and $\mathbb{Z} _2 $

What does a homomorphism between the free product $\mathbb{Z} \star \mathbb{Z} $ and $\mathbb{Z}_2 $ look like? I'm having trouble trying to do anything with this.
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Why does no non-trivial reduced words in $H_1 \backslash \{e\} \sqcup H_2 \backslash \{e\}$ imply that $\left< H_1, H_2 \right> = H_1 \ast H_2$?

I am currently studying combinatorial group theory and am trying to prove the ping pong lemma. Many of the proofs I have come across seem to use a result of the following flavour. Let $G$ be a group ...
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Showing that the free group of a disjoint union is isomorphic to the free product of the corresponding free groups

P. Aluffi's "Algebra: Chapter $\it 0$", exercise II.$5.8$. Still more generally, prove that $F(A\amalg B)=F(A)*F(B)$ and that $F^{ab}(A\amalg B)=F^{ab}(A)\oplus F^{ab}(B)$ for all sets $A,B$...
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Quotient of amalgamated free product by normal subgroup

I was given the following question in my homework: Given $Γ =G_1∗_HG_2$, suppose that $H$ is a normal subgroup in both $G_1$ and $G_2$. (a) Show that $H$ is then a normal subgroup in $Γ$. (b) Show ...
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$\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z}$ is an index two subgroup of $\mathbb{Z}\ast\mathbb{Z}$

I want to prove $\mathbb{Z}\ast\mathbb{Z}\ast\mathbb{Z}$ is an index two subgroup of $\mathbb{Z}\ast\mathbb{Z}$. Can I use covering map to prove this? Any ideas and suggestions are greatly appreciated....
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1answer
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Understanding the free product amalgamation with an example

I am trying to understand the $\ast -$product of two groups, I think I have managed to understand the free product of two arbitrary groups, but I am having problems when I consider $\ast_G$. The ...
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Intuition of characteristic property of the free group

Here is a theorem about characteristic property of the free group: Theorem (Lee TM). Let $S$ be a set. For any group $H$ and any map $f:S\to H$, there exists a unique homomorphism $g:F(S)\to H$ ...
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Is this the way to embed $G_1$, $G_2$ into the amalgamated product $G_1*_HG_2$?

Let $G_1,G_2,H$ be groups such that there are injective homomorphisms $f_i\colon H\to G_i$ for $i=1,2$. We know that there are injective homomorphisms $\iota_i \colon G_i\to G_1*G_2$, where $G_1*G_2$ ...
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1answer
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Free groups and free product of groups are isomorphic

I saw the following conclusion from a reference book: Let $G=*_{i\in I}G_i$ be free product of $G_i$, where each $G_i$ is a group. If $I=\{1,\cdots, n\}$, each $G_i=\Bbb Z$, then $G$ is isomorphic ...
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What is the correct name for a "product function" on a monoid?

Let $W$ be a monoid. A function $f\colon W\rightarrow W$ is a "product function" if $f(w)$ is a product of constants in $W$ and positive integer powers of $w$. It could also be called a "non-...
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Free Probability and Vandermonde

There is a paper entitled: "Random Vandermonde Matrices-Part I: Fundamental results" that I wish to understand some math handling steps. The manuscript can be downloaded from https://documents.epfl.ch/...
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Let $x^2=y^2=1$ and $xy\neq yx$. There are $\binom{2n}{n}$ expressions of length $2n$ in $x$ and $y$ that are equal to $1$.

This question is motivated by this link. The statement is as follows. (Edit: Even if there are already two great answers, I would love to have a couple more answers. Especially, I would like to see ...
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Reduced decompositions of elements of an amalgamated sum of monoids

Let $(M_i)_{i\in I}$ be a family of monoids, $A$ a monoid and $(h_i:A\rightarrow M_i)_{i\in I}$ a family of homomorphisms. Let $M$ be the sum of the family $(M_i)_{i\in I}$ amalgamated by $A$. Let $\...
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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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1answer
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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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1answer
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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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1answer
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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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1answer
222 views

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 ...
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About finitely presented algebras and free product

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, ...
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190 views

What's the free product of $\Bbb Z/3\Bbb Z *\Bbb Z/3\Bbb Z $?

What's the free product of $\Bbb Z/3\Bbb Z *\Bbb Z/3\Bbb Z $? Let $G=H=\Bbb Z/3\Bbb Z *\Bbb Z/3\Bbb Z =\langle g\in\{0,1,2\}\mid 3g\equiv0\pmod 3\rangle$ Then I have that the free product of groups $...
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1answer
247 views

Free product with amalgamation vs pushout [duplicate]

As in title, in terms of group theory (I'm not familiar with category-theoretic terms), question comes from algebraic topology but seems to be of general interest. (Other questions on MSE touch on the ...
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3answers
441 views

Difference between "generating set" and free product?

Let $G$ and $H$ be free groups and $g \in G$. Is there a difference between $\langle H, G\rangle$ and the free product $H*G$?. In particular is $\langle H ,g \rangle = H * \langle g \rangle$?