All Questions
Tagged with free-groups group-homomorphism
19
questions
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extension condition for free abelian groups
if $G$ is a free abelian group with basis {${a_\alpha}$} then given the elements {${y_\alpha}$} of an abelian group $H$, there are homomorphisms $h_\alpha : G_\alpha \to H$ such that $h(a_\alpha)=y_\...
- 491
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1
answer
45
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A question related to an induced homomorphism between two groups
Suppose $X$ is obtained by gluing two tori at a single point and let $r:\sum_2\to X$ be the retraction given by collapsing a circle around the middle of $\sum_2$ (surface of genus $2$) to a single ...
- 520
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1
answer
118
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Does a free group $F$ of finite rank $n$ have finitely many retracts (as a subgroup)?
A subgroup $H$ of a group $G$ is called a retract of $G$ if there exists an epimorphism $r:G\to H$ such that $r(h)=h$ for all $h\in H$.
Does a free group $F$ of finite rank $n$ have finitely many ...
- 2,783
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1
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147
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The endomorphism structure on a free group thinks it's a "ring" without abelian $+$, but there's no way it could be a ring.
Let $G$ be a free group on a finite alphabet $A$, and $R = \text{End}(G)$. Then we can concatenate endmorphisms pointwise to get another one. But endomorphisms preserve concatenation so this must ...
1
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1
answer
71
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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
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1
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128
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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
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1
answer
147
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Show that the kernel of a homomorphism from $G_1*G_2$ to $G_1\times G_2$ is free on $S$
Show that the kernel of $\omega: G_1*G_2 \to G_1 \times G_2$ given by $\omega(g_1g_2)= \iota_1(g_1) \cdot \iota_2(g_2) = (g_1,g_2)$ is free on $S=\{ gg'g^{-1}g'^{-1}, 1\neq g \in q_1(G_1), 1 \neq g'\...
- 127
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1
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452
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Hom( , ) and Hom_F( , ) and tensor product
My professor said in his lecture, "For Abelian groups $A$, $B$, $C$ and field $F$ $\operatorname{Hom}(A,B)=\operatorname{Hom}_\mathbb{Z} (A, B)$ but $\operatorname{Hom}(C, F) \neq \operatorname{...
- 113
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2
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292
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Epimorphism from free product to direct product of groups.
Let $G_1$ and $G_2$ be two non abelian groups and $G_1 * G_2$ be the free product of these two groups. Can we define an epimorphism from $G_1 *G_2$ to the direct product $G_1 \times G_2$ of these ...
- 1,954
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43
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An isomorphism onto the additive group $\mathbb{Z}\times\mathbb{Z}$
Let $I=\{\alpha,\beta\}$ such that $\alpha\ne\beta$. Let $F(I)$ be the free group constructed on $I$ and $\phi_\alpha,\phi_\beta$ be the canonical injections of $\mathbb{Z}$ into $F(I)$. Write $r=\...
1
vote
1
answer
519
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Free groups, generators and group homomorphisms
Let $F_2$ be the free group with generators $x_1$,$x_2$, and let $F_3$ be the free group with generators $y_1$,$y_2$,$y_3$.
We define a group homomorphism $\phi:F_3\rightarrow F_2$ by $\phi(y_1):=x_1^...
- 711
1
vote
1
answer
79
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At most one homomorphism between two groups
Let $S$ be a set, and $F$, $G$ be groups.
Let $f: S \rightarrow F$ and $g: S \rightarrow G$ be functions.
I want to prove the following:
If $f(S)$ generates $F$, then there exists at most one ...
- 1,191
2
votes
1
answer
261
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Number of homomorphism between a free group $F(S)$ and a group $G$
Let $F(S)$ be a free group with finite rank, $G$ a group with order $n$.
I want to know if the universal property can help determine the number of homomorphisms $F(S)\rightarrow G$. Say $H$ is the ...
3
votes
2
answers
156
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image of homomorphism contains free group, then so does domain
This question is a question which I had on my abstract algebra exam, I could not solve it:
True or false: suppose $f:G \to H$ is a group homomomorphism and assume that $\operatorname{Im}(f)$ ...
- 4,488
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1
answer
47
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Finitely Generated Free Group to Finitely Generated Free Monoid
Let $F_n$ be the free group on $n$ generators $u_1,...,u_n$ and $M_n$ the free monoid on $n$ generators $v_1,...,v_n$. Would $u_i \to v_i$ and $u_i^{-1} \mapsto v_i$ extend to a well-defined map that ...
- 8,040