All Questions
Tagged with free-groups free-product
41
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
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
3
votes
1
answer
116
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 ...
- 8,040
2
votes
1
answer
189
views
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 ...
- 63
2
votes
2
answers
212
views
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 ...
- 592
4
votes
1
answer
102
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 ...
- 619
2
votes
1
answer
113
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 ...
- 619
0
votes
1
answer
146
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 ...
- 711
0
votes
1
answer
136
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 $...
- 2,479
4
votes
2
answers
242
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 ...
- 54.6k
1
vote
0
answers
112
views
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 ...
- 2,663
6
votes
1
answer
127
views
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 ...
- 563
3
votes
2
answers
189
views
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 ...
- 563
6
votes
1
answer
252
views
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\...
- 2,307
5
votes
1
answer
134
views
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, ...
- 2,663