All Questions
Tagged with free-groups group-isomorphism
24
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
1
vote
0
answers
150
views
Normal subgroup of fundamental group of Klein Bottle
Let $K$ the Klein Bottle and $\pi_1(K) = \langle a,b \mid b a b a^{-1} \rangle $ be the fundamental group of the Klein bottle. Observe that $\langle b \rangle $ is a normal subgroup of $\pi_1(K)$, ...
- 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
0
votes
1
answer
131
views
Showing $F_X \cong F_Y\implies |X| = |Y|$ [duplicate]
Lately I've been studying free groups, I'm a layman on the subject but I came across a step in the demonstration that I couldn't move forward. I know the question seems to be good but: If $F_X \cong ...
- 3
0
votes
0
answers
68
views
Isomorphism of quotient groups: if $G_1 \cong G_2$, then $G_1/K_1 \cong G_2/K_2$?
Well, I have a little doubt. If $G_1$ and $G_2$ are free groups in $H_1$ and $H_2$ respectively. If we have $\langle x^2 : x \in G_1 \rangle = K_1 \lhd G_1$ and $\langle y^2 : y \in G_2 \rangle = K_2 \...
- 67
5
votes
1
answer
135
views
Morphism of free groups that induces isomorphism on abelianizations
I came up with the following question, that I'm not able to prove or disprove.
Let $\phi: F_I \to F_J$ a morphism between the free groups generated by the sets $I$ and $J$. This induces a morphism ...
- 63
-2
votes
1
answer
47
views
If $H , K \trianglelefteq F_2$ with $F_2/H\cong F_2/K$ then $H=K$ [closed]
This is probably a basic fact of group theory but I am not able to prove it:
Let $F_2$ be the free group generated by 2 elements and $H,K$ be two normal subgroups of $F_2$. If $F_2/H$ is isomorphic ...
- 861
3
votes
1
answer
178
views
Can the direct product of nontrivial groups $A\times B$ ever be isomorphic to a free group?
Can the direct product of nontrivial groups $A\times B$ ever be isomorphic to a free group?
Let's say that for two groups $A$ and $B$ we have that $A\times B \cong F_n$ where $F_n$ is the free group ...
- 2,454
2
votes
1
answer
45
views
Is $\langle S \rangle / \operatorname{nc}(T) \simeq \langle S \setminus T \rangle$, where $T \subseteq S$?
Let $S$ be a set and $\langle S \rangle$ denote the free group generated by the set $S$.
If we take a subset $T$ of $S$ and consider the quotient group $\langle S \rangle / \operatorname{nc}(T)$, ...
- 2,663
3
votes
0
answers
75
views
Equivalence of two elements of the free group under automorphisms?
I have two elements of the form
$$
w = x_1^{a_1} x_2^{a_2} x_3^{a_3} x_4^{a_4} x_5^{a_5} x_6^{a_6}
$$
and
$$
w' = x_1^{b_1} x_2^{b_2} x_3^{b_3} x_4^{b_4} x_5^{b_5} x_6^{b_6}
$$
for integers $a_i$ and $...
- 5,311
4
votes
2
answers
283
views
Why the group $\langle x,y\mid x^3, y^3, yxyxy\rangle$ is not trivial?
This comes from Artin Second Edition, page 219. Artin defined $G=\langle x,y\mid x^3, y^3, yxyxy\rangle$ , and uses the Todd-Coxeter Algorithm to show that the subgroup $h=\langle y\rangle$ has ...
- 127
1
vote
0
answers
65
views
Show that two presentations are isomorphic
It is known that if there is a non-abelian group of order $pq$, then it must be the case that $q\mid p-1$ and this group is isomorphic to $\langle a,b:a^p=b^q=1,ab=ba^u\rangle $ wherein $u$ is of ...
- 1,063
6
votes
2
answers
199
views
Find number of normal subgroups in $F_3$ such that its factor is isomorphic to a given Abelian group
On the upcoming test I will be given a problem of type:
Find all normal subgroups $H$ in $F_n$ such that $F_n/H \cong G$.
Here $n$ is a small integer, likely 2 or 3, and $G$ is an Abelian group ...
- 446
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
3
votes
0
answers
109
views
What are some examples of groups each isomorphic to a subgroup of each other, although not themselves isomorphic?
Do there exist groups $G$ and $H$ for which $G$ is isomorphic to a subgroup of $H$ and $H$ is isomorphic to a subgroup of $G$, but in fact $G$ is not isomorphic to $H$?
I know that $G = F_2$ and $H = ...
- 524