All Questions
Tagged with free-groups solution-verification
30
questions
5
votes
3
answers
1k
views
Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same?
I have read on different sources that it is not possible to give a simple proof that "two free groups are isomorphic if and only if they have the same rank" using only what "a student ...
- 69
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
98
views
$F_X \cong F_Y \Rightarrow |X| = |Y|$ where is the mistake in this proof
Statement.
Let $F_X, F_Y$ free groups over $X,Y$ respectively. Suppose there is an isomorphism $\phi: F_X \cong F_Y$. Then $|X|= |Y|$.
My proof. Let $x \in F_X$ be a word of length one, this is, $x = ...
- 315
3
votes
0
answers
134
views
is $F_2$ a subgroup of any other $F_n$ for $n\geq 2$
This is more to check an argument
Since $F_2$ is a group generated by words of two generators, call them $\{a,b\}$
now every other $F_n$ (provided $n\geq 2$) will be all free words of more generators $...
- 746
1
vote
2
answers
130
views
$\langle a,b\mid ab=1\rangle\cong \mathbb{Z}$
I'm new to free groups and presentation of groups, and I am having some problems with some basic facts:
Let $\langle a,b\mid ab=1\rangle$ be a group presentation. I want to show that $\langle a,b\mid ...
- 763
2
votes
1
answer
531
views
Show that there is a unique ring homomorphism from $\mathbb{Z}$ into any ring.
$\newcommand{\Z}{\mathbb{Z}}$
For my purposes, a ring is always assumed to be a ring with identity and so because of this my working definition of a ring homomorphism requires that $\phi(1_S) = 1_R$ ...
- 2,673
2
votes
0
answers
132
views
Universal Property of the Free Abelian Group on $S$.
This problem is from Dummit and Foote, 6.3.11.
Problem: Let $S$ be a set. The group with presentation $(S,R)$ with $R = \{[s,t] \mid s,t \in S\}$ is called the free-abelian group on $S$ denoted $A(S)$....
- 2,673
0
votes
1
answer
94
views
If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.
If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.
Definition. A path-connected space whose fundamental group is isomorphic to a given group $G$ and which has ...
- 2,454
1
vote
1
answer
71
views
Draw a covering of $S^1 \vee S^1$ whose fundamental group is isomorphic to $\ker \Phi : F_2 \to \mathbb Z$ with $a\mapsto 2, b\mapsto 3$
$\newcommand{\Z}{\mathbb Z}$
Let $K\leq F_2 = \langle a,b\rangle$ be the kernel of the map $\Phi: F_2 \to \Z$ sending $a$ to $2$ and $b$ to $3$. Draw a cover of $S^1 \vee S^1$ whose fundamental group ...
- 2,454
0
votes
1
answer
128
views
Any injective homomorphism $F_n\to F_n$ with image of finite index is bijective ($n\geq 2$)
$\DeclareMathOperator{\im}{Im}$
Let $\Phi: F_n\to F_n$ be an injective homomorphism between free groups with $n\geq 2$ and $\im{\Phi}$ having finite index. Prove that $\Phi$ is bijective, i.e. $\im{\...
- 2,454
1
vote
0
answers
82
views
Free left $R$-module can be endowed with $(R,R)$-bimodule structure. Basis-dependent definition?
Let $R$ be a non-commutative ring. If we have a free left $R$-module $F$, we can consider a basis $\{e_i\}_{i\in I}$ in $F$ and endow $F$ with a $(R,R)$-bimodule structure in the following way: For a ...
1
vote
1
answer
200
views
A group is locally free exactly when its finitely generated subgroups are free
This is Exercise 6.1.9 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search for "locally free finitely generated" in the group theory tag, ...
- 45.7k
4
votes
1
answer
605
views
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 $A,B$...
- 16.2k
0
votes
1
answer
267
views
Showing this quotient group is isomorphic to $\mathbb{Z}_2\times{}\mathbb{Z}_2$
How do I show that $\mathbb{Z}_4\times{}\mathbb{Z}_2/K$ is isomorphic to $\mathbb{Z}_2\times{}\mathbb{Z}_2$, where $K=\langle(2,0)\rangle$.
I'm getting confused with the details involved here, I will ...
- 1,285
0
votes
0
answers
370
views
Products in the free group are associative
In Aluffi's Chapter 0 there is an exercise to show that the product in the concrete construction of a free group is associative by showing that the reduction of a word is independent of the order in ...
