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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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The free product has the direct product as a factor group. What's the corresponding normal subgroup?

Let $G$ and $H$ be groups. Consider the free product $G * H$ and the direct product $G \times H$. There is a particular way of identifying $G \times H$ as a factor group of $G * H$. Namely, the ...
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Finding free subgroup $F_2$ in the free product $\frac{\mathbb{Z}}{5\mathbb{Z}} * \frac{\mathbb{Z}}{6\mathbb{Z}}$

Is there any free group isomorphic to $F_2$ contained in the free product group $\frac{\mathbb{Z}}{5 \mathbb{Z}}* \frac{\mathbb{Z}}{6 \mathbb{Z}}?$ Let $\frac{\mathbb{Z}}{5\mathbb{Z}}= \langle a \mid ...
jay sri krishna's user avatar
-1 votes
1 answer
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Coproduct of free objects [duplicate]

In the category of groups, the coproduct is free product, and for any $A,B$ sets: the coproduct of $F(A),F(B)$ is $F(A\sqcup B)$. Does this property hold in any other categories? What is the ...
rutruttt's user avatar
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3 votes
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Munkres' Topology theorem 68.7

Theorem 68.7 Let $ G = G_1 * G_2 $. Let $ N_i $ be a normal subgroup of $ G_i $, for $ i = 1, 2 $. If $ N $ is the least normal subgroup of $ G $ that contains $ N_1 $ and $ N_2 $, then $$ G/N \simeq \...
Davood Karimi's user avatar
3 votes
2 answers
250 views

Munkres lemma 68.5

I'm reading Munkres Topology and I'm stuck in lemma 68.5 as you can see he uses the theorem 68.4 in order to imply that there is a isomorphism between $G$ and $G'$, but in order for this theorem to be ...
Davood Karimi's user avatar
1 vote
2 answers
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the free product of two presentations is isomorphic to a third presentation using UP of free product.

Here is the question that I want an answer to it using commutative diagrams (as small number of them as possible): Prove that the free product of $ \langle g_1, \dots ,g_m | r_1, \dots ,r_n \rangle$ ...
Intuition's user avatar
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How will the map be described? [closed]

Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\...
Hope's user avatar
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Free Product of interpolated, free group factors

Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{1/\...
Jayakumar Ravindran's user avatar
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How can I decide whether two groups defined by finite presentations are (or not) isomorphic?

I have the groups $G_1,G_2$ with presentations $$G_1 = \langle x,y : (y^2x)^2 = x^2, (x^2 y )^2 = y^{-2} \rangle = \langle x,y : x^{-1}y^2 x = y^{-2}, yx^2y^3 = x^{-2} \rangle \\ G_2 = \langle x,y : (...
Adrian's user avatar
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1 answer
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What is the operation involved for the words of free product of groups?

I am trying to get an intuitive mental picture of what the free product of two groups represents. From what I understood, the free product $G\ast H$ is the group whose elements are the reduced words ...
Vincent's user avatar
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3 votes
1 answer
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Left adjoint to forgetful functor from groups to groupoids, generalizing injective inclusions to free product of groups

Is there a left adjoint $F$ to the "forgetful" inclusion functor $U$ from the category of groups (interpreted as groupoids with one object $*$) to the category of groupoids? If so, then ...
I Eat Groups's user avatar
1 vote
1 answer
160 views

Group action of Free Product Group

Suppose I have two groups G and H, and K is their free product, K = $G*H$ Suppose G acts on a set X with action $\phi$ and H acts on X by action $\psi$, then what is the action of K on X. I think the ...
text_math's user avatar
2 votes
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A group equation and free product

Definition: Given $w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x})\in F_m ∗ F_n$ and a group $G$. The system of equations $\{w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x}) \}$ is solvable in G ...
pharazphazel's user avatar
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1 answer
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Is the free product contained in amalgamated product?

I'm trying to understand the subgroups of finite index of amalgamated products in some particular cases and it arises the following question: Suppose that $G = A \ast_C B$ where $A$ and $B$ are ...
Greg's user avatar
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If a group $G$ is $(2,3,t)$-generated, then $G$ is a factor of the modular group which is isomorphic to $\mathbb{Z}_2 * \mathbb{Z}_3$

If a group $G$ is $(2,3,t)$-generated, where $t$ is an odd divisor of $|G|$ then $G$ is a factor of the modular group which is isomorphic to $\mathbb{Z}_2 * \mathbb{Z}_3$. From assumptions, let $x$ ...
user avatar
3 votes
2 answers
121 views

What is the index of $A\ast(ba)B(ba)^{-1}$ in $A\ast B$?

A group is called co-Hopfian if it is not isomorphic to a proper subgroup of itself, while it is called finitely co-Hopfian if it is not isomorphic to a proper finite index subgroup of itself. A co-...
Michael Albanese's user avatar
2 votes
1 answer
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System morphisms forming an amalgamated free product can be supposed to be injective

This is exercise 1.1(1) of Serre's Arbres, amalgames, SL$_2$. I explain what I mean by the title of the question in the last paragraph. Let $f_1:A\to G_1$ and $f_2:A\to G_2$ be two group homomorphisms ...
Marc-André Brochu's user avatar
3 votes
1 answer
107 views

Normalizer of Factors in an Amalgamated Free Product

I am reading through this proof that if $H$ is a non-trivial group, then the normalizer of $H$ in the free product $G:=H \ast K$ is equals $H$ (i.e., is trivial). Most of the proof seems to generalize ...
user193319's user avatar
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How to show direct sum of free product of groups is not isomorphic to free product of direct sum of groups?

I guess that $(\mathbb Z \times \mathbb Z) *\mathbb Z$ is not isomorphic to $\mathbb Z \times (\mathbb Z *\mathbb Z)$ since there's no such associative law. What I try: we can write these two groups ...
iefjkfdhfure's user avatar
4 votes
2 answers
201 views

How to prove $\mathbb Z^2 \ast \mathbb Z$ is quasi isometric to $\mathbb Z^2 \ast \mathbb Z^2$?

How to prove $\mathbb Z^2 \ast \mathbb Z$ is quasi isometric to $\mathbb Z^2 \ast \mathbb Z^2$? Here $\ast$ is the free product of groups. I am thinking of proving they are commensurable. In other ...
V SMASH's user avatar
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2 votes
1 answer
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Why doesn't the infinite dihedral group contain a free subgroup of rank 2?

Our professor just told us that $D_{\infty}$ is "too small" whatever that means. Can someone prove this statement and give some reason as to why this statement holds true but doesn't for ...
shekh's user avatar
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4 votes
1 answer
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Proof of the Wirtinger Presentation using Van Kampen Theorem

I have some difficulties understanding a proof of the Wirtinger presentation using the Van Kampen theorem, found in John Stiwell's "Classical Topology and Combinatorial Group Theory". I ...
Arthur Filippi's user avatar
1 vote
1 answer
116 views

What is the free product with amalgamation with the trivial group?

I am working on algebraic topology. I am trying to prove the Wirtinger presentation using the Van Kampen theorem. However, I have some difficulties understanding the concept of free product with ...
Arthur Filippi's user avatar
2 votes
0 answers
95 views

HNN-extension and Centralizer

I am currently studying the book of Graham Higman and Elizabeth Scott, The Existentially Closed Groups, London Mathematical Society Monographs New Series, Clarendon Press Oxford, 1988. In the Section ...
mathmehmet's user avatar
1 vote
1 answer
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Are the only finite orders of elements of this group in $\{p^k\mid k\in\Bbb N\cup\{0\}\}?$

Let $p$ be prime. Consider the group $G$ given by the presentation $$P=\langle a,b,\{ x_m\mid m\in\Bbb N\}\mid a^p, \{x_m^{p^m}, x_m=b^mab^{-m}\mid m\in\Bbb N\}\rangle.$$ My question is two-fold: Are ...
Shaun's user avatar
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2 votes
2 answers
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Computation of Amalgamated Product $\mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6.$

I'm trying to compute a amalgamated product $\mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6$. Let $\mathbb{Z}_4= \langle a\mid a^4 =1\rangle$ and $\mathbb{Z}_6 = \langle b\mid b^6 =1\rangle $, be a ...
Horned Sphere's user avatar
4 votes
0 answers
106 views

Understanding free product groups

I am trying to understand free product groups and I considered the group $G=\langle a,b|a^2b^{-3}\rangle$. As in this group $a^2=b^3$, it seems like every element of G can be written as $b^{n_1}ab^{...
DagunDagun's user avatar
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Coalgebra structure on $T(V)=k\oplus V \oplus V\otimes V \oplus V^{\otimes 3}\oplus \cdots $

In the wiki page:https://en.wikipedia.org/wiki/Cofree_coalgebra They discuss two coalgebra structures on $T(V)$ I dropped the tensor between $v_1\otimes \cdots\otimes v_n$, the two coproducts: $$\...
IrbidMath's user avatar
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4 votes
1 answer
101 views

Order of $xy$ in various quotients of the free product $G_1 * G_2$, where $x \in G_1, y \in G_2$ are nontrivial

Say we have two arbitrary nontrivial groups $G_1$ and $G_2$, and some arbitrary nontrivial elements $x \in G_1, y \in G_2$. Then it is known that the order of $xy$ in the free product $G_1 * G_2$ is ...
I Eat Groups's user avatar
2 votes
1 answer
110 views

Unique factorization of free products of groups satisfying descending chain condition

I am self-studying group theory, and proving Exercise 11.61 of Rotman's An Introduction to the Theory of Groups, on free products: Let $A_1, \ldots, A_n, B_1, \ldots, B_m$ be indecomposable groups ...
I Eat Groups's user avatar
3 votes
1 answer
93 views

Group Action and Free Product of Groups

I am struggling on the problem related to the free product of groups. Let $G$ be a group acting on a set $X$ and let $H$ be a group generated by two subgroups $G_1, G_2$ of $G$ with $\lvert G_1 \...
Alex Lee's user avatar
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0 answers
76 views

Classifying index 2 subgroups of $\langle a, b, c \mid ab = ba, c^2 = 1 \rangle$ up to isomorphism

Let $G := \langle a, b, c \mid ab = ba, c^2 = 1 \rangle$. By considering non-trivial homomorphisms $\phi : G \rightarrow \mathbb Z/2\mathbb Z$, there are exactly 7 subgroups of $G$ given by kernels of ...
Luke's user avatar
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4 votes
1 answer
182 views

Is $G*G \cong G$ where $G$ is the fundamental group of the Hawaiian earring?

Let $X$ be the Hawaiian earring and let $x_0$ be the point $(0,0)$. Consider the wedge sum $Y:=X\vee X=X\coprod X/x_0\sim x_0$. Then using $\Bbb N=\{1,3,5,\dots\}\cup \{2,4,6,\dots\}$, $X$ is ...
user302934's user avatar
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0 votes
1 answer
34 views

Generalizing an argument of free products: If $f\circ\alpha, f\in \mathrm{Hom}(A,B)$ and $|\mathrm{Hom}(A,B)|=1$, do we know $\alpha = \mathrm{Id}?$

I'm studying the free product of two groups $A$ and $B$. In order to show that there is a unique free product, one considers the following: Suppose that $G, X$ are two free products of $A$ and $B$. ...
Aericura's user avatar
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1 answer
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Why is the random walk on the modular group transient?

I have been reading about random walks on Cayley graphs of groups lately and stumbled across the walk on the modular group $\mathbb{Z}/(2\mathbb{Z}) * \mathbb{Z}/(3\mathbb{Z})$, where $*$ denotes the ...
squareandroot's user avatar
2 votes
2 answers
77 views

Help with amalgated product (calculating $\pi_1(\mathbb{R}P^2)$)

I’m trying to calculate $\pi_1(\mathbb{R}P^2)$ using Van Kampen’s theorem. After choosing the two open sets (which I call $U$ and $V$) according to my lecture notes, I get $$\pi_1(\mathbb{R}P^2)=\pi_1(...
Yamido's user avatar
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4 votes
2 answers
161 views

Finite index subgroup isomorphic to $\Bbb{Z}$ inside the free product $\Bbb{Z}_{2}*\Bbb{Z}_{2}$

Let $\Bbb{Z_{2}*Z_{2}}$ be the free product then I want to prove that there is a finite indexed subgroup isomorphic to $\Bbb{Z}$. My attempt: We denote the free product by $\langle a\rangle*\langle b\...
Dovahkiin's user avatar
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11 votes
1 answer
250 views

Can a free product of groups be co-Hopfian?

A group $G$ is called co-Hopfian if it is not isomorphic to a proper subgroup of itself; equivalently, every injective group homomorphism $\varphi : G \to G$ is surjective and hence an isomorphism. ...
Michael Albanese's user avatar
0 votes
1 answer
30 views

Summands of a free product with amalgamation

I'm currently reading Freedman's paper on the Mobius energy of knots. In the proof of Theorem 4.3, he constructs a cylindrical covering $N$ of a tame knot $\gamma_K$, contained a tubular neighborhood $...
maxematician's user avatar
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3 votes
1 answer
248 views

Understanding quotient group in free product (van Kampen theorem)

I have never studied algebra thoroughly, so I do not have any experience with groups and no mindset that handles them naturally. For example, yesterday I have been looking into examples of quotient ...
SBF's user avatar
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1 vote
0 answers
193 views

Injectivity of maps in a free product group with amalgamation

Given countable groups $H, G_1$ and $G_2$ and group homomorphisms $\theta_i: H \rightarrow G_i$, we can construct the free group with amalgamation $(G_1 \ast G_2)/N$ where $N$ is the normal subgroup ...
abcdef's user avatar
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5 votes
3 answers
182 views

Trying to understand the differences between $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$

I’m trying to understand the differences between free products and direct products with an example: $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$. If I understand correctly, the ...
lanero's user avatar
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0 votes
1 answer
134 views

Ping-pong lemma assumptions

The ping-pong-lemma for subroups is often stated with the following assumptions, e.g. taking this one from Wikipedia: Let G be a group acting on a set $X$ and let $H_1, H_2, ..., H_k$ be subgroups of ...
Joachim Breitner's user avatar
1 vote
1 answer
613 views

Is the infinite dihedral group isomorphic to $\mathbb{Z}*\mathbb{Z}$?

Let $a,b$ denote the generators of the copies of $\mathbb{Z}_2$ in the free product $\mathbb{Z}_2*\mathbb{Z}_2$. The infinite dihedral group is described by: $$D_{\infty} = \; \left<r,s \;|\; srs=r^...
Adam_math's user avatar
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0 votes
1 answer
131 views

Is $\Bbb Z^n$ isomorphic to $\Bbb Z*\Bbb Z*\Bbb Z ...$, $n$ times?

Let $\mathbb{Z}^n$ be the group formed by the external direct product of $\mathbb{Z}$ taken $n$ times, and let $A_n$ be the group formed by taking the free product of $\mathbb{Z}$ $n$ times. Then, is $...
Anon's user avatar
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5 votes
2 answers
108 views

Show that $(\mathbb{Z}*\mathbb{Z})/[\mathbb{Z}*\mathbb{Z},\mathbb{Z}*\mathbb{Z}]\cong \mathbb{Z}\oplus\mathbb{Z}$

Im trying to show that $$(\mathbb{Z}*\mathbb{Z)}/[\mathbb{Z}*\mathbb{Z},\mathbb{Z}*\mathbb{Z}]\cong \mathbb{Z}\times\mathbb{Z}$$ and my first thought was to use the first isomorphism theorem. I've ...
lspacebarl's user avatar
0 votes
1 answer
80 views

Generators give quotients of free products of groups?

I have some confusion here. I have this idea that: If $\Gamma$ is generated by (all of) the elements of subgroups $G_1,\dots,G_k$, then $\Gamma=\langle G_1,\dots, G_k\rangle$ is a quotient of the ...
JP McCarthy's user avatar
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4 votes
2 answers
227 views

A certain free product of groups is virtually torsion-free

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 ...
Alex Youcis's user avatar
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3 votes
0 answers
118 views

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 ...
arnett's user avatar
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3 votes
1 answer
279 views

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 ...
Anakhand's user avatar
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