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
167
questions
2
votes
0
answers
78
views
Free product contains the free product of itself with a free group.
I saw this answer and I'm thinking if with this idea we can show that $G_1 \ast G_2 \ast F$ (where $F$ is a free group of countable rank) embedds in $G_1 \ast G_2$ if $G_1$ or $G_2$ has cardinality at ...
- 422
0
votes
0
answers
35
views
Infinite coproduct in a category of groups [duplicate]
Let $\mathbf{Grp}$ be a category of groups.
Then, we know that there exists a coproduct $\bigsqcup_{I\in I} G_i$ for a family of groups $\{G_i\}_{I\in I}$ in $\mathbf{Grp}$ when $I$ is a finite set.
...
- 1,934
1
vote
0
answers
41
views
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 ...
- 2,049
9
votes
2
answers
134
views
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 ...
- 435
-1
votes
1
answer
48
views
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 ...
- 111
3
votes
0
answers
83
views
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 $$ G/N \simeq \...
- 491
3
votes
2
answers
255
views
Munkres lemma 68.5
I'm reading Munkres Topology and I'm stuck in lemma 68.5 as you can see he uses the theorem 68.4 in order to imply that there is a isomorphism between $G$ and $G'$, but in order for this theorem to be ...
- 491
1
vote
2
answers
94
views
the free product of two presentations is isomorphic to a third presentation using UP of free product.
Here is the question that I want an answer to it using commutative diagrams (as small number of them as possible):
Prove that the free product of $ \langle g_1, \dots ,g_m | r_1, \dots ,r_n \rangle$ ...
- 3,127
-1
votes
1
answer
65
views
How will the map be described? [closed]
Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\...
- 95
1
vote
0
answers
54
views
Free Product of interpolated, free group factors
Let $L(\mathbb F_2)$ be the group von Neumann algebra of the free group on two generators. The interpolated free group factors of Dykema and Radulescu are defined as $L(\mathbb F_r)=L(\mathbb F_2)^{1/\...
4
votes
0
answers
119
views
How can I decide whether two groups defined by finite presentations are (or not) isomorphic?
I have the groups $G_1,G_2$ with presentations $$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 : (...
- 61
1
vote
1
answer
40
views
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 ...
- 1,445
3
votes
1
answer
68
views
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 ...
- 619
1
vote
1
answer
176
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 ...
- 23
2
votes
0
answers
64
views
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 ...
- 47