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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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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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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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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 $\langle H ,g \rangle = H * \langle g \rangle$?
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Prove that $\Bbb Z\oplus \Bbb Z_2$ is not isomorphic to $\Bbb Z∗\Bbb Z_2$.

I need to prove that $\Bbb Z\oplus \Bbb Z_2$ is not homeomorphic to the free product $\Bbb Z∗\Bbb Z_2$. I know that $\Bbb Z\oplus \Bbb Z_2$ is abelian while the free product $\Bbb Z∗\Bbb Z_2$ is not ...
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What are the elements of $\Bbb Z\oplus \Bbb Z_2$ and $\Bbb Z∗\Bbb Z_2?$ [closed]

What are the elements of the following groups? (1) $\Bbb Z\oplus \Bbb Z_2$ , (2) $\Bbb Z∗\Bbb Z_2$ Thanks in advance.
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Definition of free generators of $\mathbb{Z}*\dotsb*\mathbb{Z}*\mathbb{Z}_2*\dotsb*\mathbb{Z}_2$

I didn't quite understand the definition of $g_1,...,g_k$ in the first paragraph of page 3 of this article: Let $0\leq q\leq d/2$ and $X=X_*=\mathbb{Z}*\dotsb\mathbb{Z}*\mathbb{Z}_2*\dotsb*\mathbb{Z}...
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Is the free product of residually finite groups always residually finite?

Suppose groups $G$ and $H$ are residually finite. Does that imply, that $G \ast H$ is residually finite? What have I tried to prove this: Suppose, $a = g_1h_1g_2h_2…g_nh_n \in G \ast H$, $g_1, .. ...
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3answers
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Does there exist a group that is both a free product and a direct product of nontrivial groups?

Do there exist such nontrivial groups $A$, $B$, $C$ and $D$, such that $A \times B \cong C \ast D$? I failed to construct any examples, so I decided to try to prove they do not exist by contradiction....
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Kernel of $G\ast H\to G\times H$ is free

Let $G$ and $H$ be groups. The identity homomorphism $G\to G$ and the trivial homomorphism $H\to G$ give a homomorphism $G\ast H\to G$ by the universal property of the coproduct. Similarly, we obtain ...
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Free subgroups of $PSL(2,\mathbb{Z})$ of index 6

There are two "natural" subgroups of $PSL(2,\mathbb{Z})\cong C_2\ast C_3$ of index 6. One is the congruence subgroup $\Gamma_0(2)$ which is the kernel of the map $PSL(2,\mathbb{Z})\to PSL(2,\mathbb{Z}/...
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project normal subgroup generated by a subgroup to its abelianization

Say $\operatorname{Ab}(G)$ is the abelianization of $G$. Let $G_1$ and $G_2$ be two groups, $G_1\times G_2$ is the free product, then $G_1$ can be viewed as a subgroup in it. $j:G_1\times G_2\...
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Showing that there is a surjective map from $\Bbb Z \ast \Bbb Z$ to $C_2 \ast C_3$ just using universal property of coproduct

I am solving Allufi chapter $0$ exercise $3.7$. There is a easy way to solve this if we know how the coproduct of $\Bbb Z \ast \Bbb Z$ and $C_2 \ast C_3$. I was wondering if there is an abstract ...
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Direct sum of non-abelian groups doesn't satisfy the universal property of direct sum.

Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum. I think I must assume that the universal property is true and then use the free ...
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Prove that doubly punctured theta space is contractible.

This is a different approach to what was suggested by Joel Pereira in my question Fundamental group of theta-space and the doubly punctured theta space in Munkres Topology Example 70.1 Munkres ...
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Fundamental group of theta-space and the doubly punctured theta space in Munkres Topology Example 70.1

Munkres Topology Example 70.1 Let X be theta-space, U = $X \setminus \{a\}$ and V = $X \setminus \{b\}$. Let $U \cap V = X \setminus \{a,b\}$ be doubly punctured theta-space where $a,b$ are interior ...
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1answer
24 views

Some preliminary concepts for Rota-Baxter algebras

I am studying Rota-Baxter Lie algebras. I do not know whether there exists the notion of free product, semi-direct product and derivation map for these types of algebras. May you introduce some papers ...
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1answer
46 views

Showing $G =\langle a,b\mid a^2=1,\ b^3=1\rangle$ is an infinite group [closed]

Consider $G =\langle a,b \mid a^2=1,\ b^3=1\rangle$. Show that $G$ is an infinite group.
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1answer
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Computing $\mathbb{Z} \ast_{\mathbb{Z}} \mathbb{Z}$

I want to compute $\mathbb{Z} \ast_{\mathbb{Z}} \mathbb{Z}$ with respect to homomorphisms: $\varphi_1:\mathbb{Z} \ni n \longmapsto an \in \mathbb{Z}$ $\varphi_2:\mathbb{Z} \ni n \longmapsto bn \in \...
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How to compute Lower Central Series by hand for this simple example

Let $G=\langle x,y,z\mid z^2=1\rangle\cong \mathbb{Z}*\mathbb{Z}*\mathbb{Z}/2$. I am interested to compute by hand (or by any other means) the quotient groups $\gamma_n/\gamma_{n+1}$ where $\gamma_n$ ...
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What is the best book to learn : Free products with amalgamation.

I'm studying groups with Abstact Algebra Grillet, in the free product part, there's a page about amalgamation with two propositions with no proofs, I will be glad if you can recommand a book where ...
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What is the product of two Gaussian random matrices?

Suppose you have two i.i.d $K \times N$ random matrices $A$ and $B$. Both matrices have complex valued entries taken from a Gaussian distribution with $\mu = 0$ and normalised variance $\sigma^2 = \...
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1answer
207 views

Royal Road to Free Groups and Free Products

This question is more about strategy, which can be used when developing group theory, then about a particular proofs. $ \newcommand{GRP}{\mathsf{GRP}} \newcommand{SET}{\mathsf{SET}} $ One way to ...
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Given homomorphisms $\phi : G \rightarrow H$ and $\psi : G \rightarrow F$, if $\phi$ is an isomorphism, is it true that $F \simeq H \ast_G F$? [closed]

Let $G$ be a group with two homomorphism $\phi \colon G \rightarrow H$ and $\psi \colon G \rightarrow F$. If $\phi$ is an isomorphism, is it true that $F \simeq H \ast_G F$, the free product with ...
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Test if group has amalgamated free product structure

I have a finitely presented group $H$. Is there an algorithm that can decide weather $H$ has any amalgamated free product structure $H\cong H_1 \star_F H_2$? Thanks.
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In the construction of the free product of C*-algebras, is the seminorm from which we quotient actually a norm?

Context Let $(A_\lambda)_{\lambda \in \Lambda}$ a family of unital C*-algebras. I am trying to see that they have a coproduct. In order to construct it, I understand that we take the coproduct of $(...
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If $N$ is the normal subgroup of $A\ast B$ generated by $A$, then $(A\ast B)/N\cong B$

Suppose that $A\ast B$ is the free product of the groups $A$ and $B$, and $N$ is the normal subgroup of $A\ast B$ generated by $A$. I wish to prove that $(A\ast B)/N\cong B$. Suppose that $h:A\ast B \...
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proof verification of an elementary result about a subgroup of a free group using a technical method

When I was proving the following exercise with a technical method (although it could be solved by a very elementary technique), I ran accross a problem. The exercise is: Let $F$ be a free group and ...
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2answers
70 views

Semidirect factors of free product of two groups

Given a group $G$ with identity element 1, a subgroup $H$, and a normal subgroup $N$ of $G$; $G$ is called the semidirect product of $N$ and $H$, written $G = N\rtimes H$ , if $G = NH$ and $H\cap N=1$...
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Finite index subgroups in Amalgamated Free products

Are there any results along the following lines: Let $\Gamma_1$ and $\Gamma_2$ be groups with respective finite index subgroups $\Gamma_0^i$ for $i=1,2$. If $\Gamma_1 \cap \Gamma_2 \leq \Gamma_0^i$ ...
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1answer
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Find a space whose fundamental group is $\mathbb Z/2 × \mathbb Z$

Find a space whose fundamental group is i) $\mathbb Z/2 × \mathbb Z$ ii) $\mathbb Z/2 ∗ \mathbb Z$ Here, $\mathbb Z$ is the set for integers. And $*$ is the free product defined as $F(G \...
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Presentations of Amalgamated Free Products of Two Groups.

Suppose we have an amalgamated free product $H\ast_LK$ of groups $H$ and $K$ with respect to a (normal) subgroup $L$ (of both $H$ and $K$), where $H\equiv\langle X\mid R\rangle$ and $K\equiv\langle Y\...
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1answer
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Is the intersection of two subgroups, defined below, always trivial?

Suppose, $G = \mathbb{Z} \ast H$, where $H$ is a torsion-free group. Suppose, $g \in G$ and $g \notin H$. Is $\langle\langle g \rangle \rangle \cap H$ always trivial? ($\ast$ stands for free product, ...
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1answer
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Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$. I think that this space is the free product $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$, but I do not know how to ...
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1answer
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Uniqueness of free product

I study Free Product of Groups in [Munkres, Topology], and I have some concerns about it. $\textbf{Basic definition}$ 1. Let $G$ be a group, and let $\{G_\alpha\}$ be a family of subgroups of $G$. ...
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Are the groups $\mathbb{Z}*(\mathbb{Z}*\mathbb{Z})$ and $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$, the same?

I was wondering if the groups $\mathbb{Z}*(\mathbb{Z}*\mathbb{Z})$ and $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$ are the same? Here $*$ operation refers to free product of two groups. Intuitively somehow ...
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1answer
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Is free product of groups always bigger that direct product?

My question is as follow: Let $\{G_\alpha\}, \alpha \in A$ be a class of groups. Is it always true that there exists a surjective homomorphism $\phi$ $$\phi:*_\alpha \,G_\alpha\, \to \prod_\alpha G_\...
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Is the map $G*H \to G \times H$ injective?

I am reading Hatcher's book on Algebraic topology. Here is a paragraph about free product of group. I think the last line should not be "a surjective homomorphism", but an injective one, since the ...
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Proof check, existence of free product

I used to fear the existence of free products and respect it, after seeing the pretty (but nontrivial) treatment in introduction to manifolds. Afterwards I was wondering why doesn't the following ...
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1answer
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$SL_2(\mathbb{Z})$ is an amalgamated product

It is known that $SL_2(\mathbb{Z})= <S,R> $ where the generators are $S=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}$ and $R=\begin{pmatrix} 1 & -1\\ 1 & 0 \end{pmatrix}$ of ...
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The relationship between semidirect product and free product

If we define $\mathbb{Z}\ltimes G$ to be $\mathbb{Z}\times G$, with multiplication as $$ (a,g)*(b,h)=(a+b,g^a\cdot h) $$ I want to show that $\mathbb{Z}\ltimes G$ is isomorphic to $\mathbb{Z}*G$(free ...
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Clarification on Lemma 68.5 of Munkres

In James R. Munkres's "Topology" he has the lemma "Lemma 68.5. Let $\{G_{\alpha}\}_{\alpha \in J}$ be a family of groups; let $G$ be a group; let $i_{\alpha}: G_{\alpha} \to G$ be a family of ...
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Is there a non-trivial group $C$ such that $A*C \cong B*C$ implies $A \cong B$?

I recently learnt that finite groups are cancellable from direct products, i.e. if $F$ is a finite group and $A\times F \cong B\times F$, then $A \cong B$. A proof can be found in this note by Hirshon....
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Commutator of free products of groups

Let $G$ be a free product of nontrivial groups $H_i$: $G=H_1\ast\cdots\ast H_n.$ Let $g_1,g_2\in G$ be such elements that: 1) $g_1,g_2\not\in H_k^g$ for all $g\in G$ and $k=1,\ \dots,\ n$; 2) $[g_1,...
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210 views

Subgroups of an amalgamated free product of index two

Let $G=A\ast_C B$ be an amalgamated free product of two infinite subgroups over their intersection. My question: Is it possible to describe in an 'explicit' way all the subgroups of $G$ of index $...
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Certain property of free products of groups

Let $G = A\ast B$ be a free product of groups $A\ne1$ and $B\ne1$ without the elements of order $2$. Suppose that $f\in G$ and $f\notin A^g,B^g$ for any $g\in G$ and let $\left \langle \left \langle f ...
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105 views

Free product has solvable word problem if its factors do

So Lyndon & Schupp's Combinatorial Group Theory states that if $A$ and $B$ are finitely generated with solvable word problem, then $A*B$ (the free product) has solvable word problem. This appears ...
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89 views

Conjugacy Classes in $\mathbb{Z}*\mathbb{Z}$? [closed]

May I know what are the conjugacy classes of the group $\mathbb{Z}*\mathbb{Z}$ (the free product of infinite cyclic groups by itself)?
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36 views

Free product of one-element groups

Suppose I had groups $G_1 = \{e_1\}$ and $G_2 = \{e_2\}$; where both groups only have a single element (the identity element). What would the free product of the two groups $G = G_1 * G_2$ be? At ...
2
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1answer
150 views

When are free products of pairs of groups isomorphic?

Suppose that $G_1$, $G_2$, $H_1$ and $H_2$ are non-trivial groups. Under what conditions do we have $$G_1 \ast H_1 \cong G_2 \ast H_2?$$ Is it the case that if $G_1 \cong G_2$ then we only have such ...
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74 views

The free group $\mathbb{F}_2$ is a subgroup of a free product

I am not so familiar with free groups so I want to study those groups. For example the free product is in general not a free group: If we consider the free product $\mathbb{Z}_2 \ast \mathbb{Z}_2 $, ...