All Questions
Tagged with free-groups abelian-groups
55
questions
2
votes
1
answer
86
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Direct sum of free abelian group and quotient of abelian group by subgroup
I'm currently studying abelian groups in Kurosh's The Theory of Groups. I'm trying to understand the proof of the theorem:
Let $B \leqq A$ be abelian groups. If $A/B \cong C$ and $C$ is a free group, ...
2
votes
1
answer
121
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Proof of the universal property of free abelian groups
Let $S$ be a set. The group with presentation $(S, R)$ , where $R = \{\ [s , t] \mid s,t\in S\ \}$ is called the free abelian group on $S$ -- denote it by $A (S)$ . Prove that $A (S)$ has the ...
- 351
-1
votes
1
answer
65
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Can you determine the order of a generator in this group presentation? [closed]
Given the following group presentation $<x,y|2x+3y=0, 5x+2y=0>$ of an Abelian group, find the order of element x.
My follow up question: Is there a way to determine the order without finding ...
- 140
0
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0
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31
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$\prod_p \mathbb{Z}/p\mathbb{Z}$ is not the direct sum of $\bigoplus_p \mathbb{Z}/p\mathbb{Z}$ and a torsion-free subgroup
While I was reading "Abelian Groups" by Fuchs $(2015)$, I encountered Example $1.2$ in the chapter on Mixed Groups, which stated the following:
Let $p_1,p_2,\dots,p_n,\dots$ denote different ...
- 69
1
vote
0
answers
114
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A subgroup $H$ of an abelian group $G$ is a direct summand iff $G/H$ is free?
Let $G$ be an abelian group and $H\le G$. Then TFAE:
$G/H$ is free.
$H$ is a direct summand of $G$.
This is how I tried to prove this theorem:
If $G/H$ is free, then $G/H$ has a basis. Let's denote ...
- 69
1
vote
2
answers
70
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Let $\Bbb Z*\Bbb Z=\langle a,b\rangle$ and $N=\{waba^{-1}b^{-1}w^{-1}:w\in\Bbb Z*\Bbb Z\}.$ Prove $\langle a,b\rangle/N$ is abelian
Let $\mathbb{Z} * \mathbb{Z} = \langle a,b \rangle$ and $$N = \left\{w a b a^{-1} b^{-1}w^{-1}: w\in \mathbb{Z} * \mathbb{Z} \right\}$$ the smallest normal subgroup that contains $\left\{ a b a^{-1} ...
- 592
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0
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61
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Understanding "formal sum" in free abelian groups
Despite reading about formal sums and especially the last comment in this post (which seems most relevant to my question) - I still feel the need to make sure I'm not missing something:
If there is a ...
- 1,791
5
votes
1
answer
135
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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
174
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Set of homomorphisms on a free abelian group is a free abelian group.
If $G$ is a free abelian group with rank $n$, I need to show that ${\rm Hom}(G,\mathbb{Z})$, set of all homomorphisms is also free abelian group of rank $n$,
My work:
Since $G$ is free abelian group ...
- 319
-1
votes
2
answers
165
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Is $\mathbb{Z_5}$a free abelian group ? Yes/No [closed]
Is $\mathbb{Z_5}$ a free abelian group ?
My attempt: I think $\mathbb{Z_5}$ is free abelian group
By the definition of free abelian group
$X$ generates $G$, and $n_1x_1 +n_2x_2 +\dots+n_rx_r=0$ ...
- 14.6k
1
vote
1
answer
188
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Lee's proof of the rank theorem for abelian groups
I am going through Prof. Lee's "Introduction to Topological Manifolds" second time through, trying to do all the exercises and problems. My question is about a proof of the rank theorem for ...
- 29.2k
1
vote
1
answer
81
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Defining a map on a subgroup of a free group
Given a set $S$, we write $G(S)$ for the free abelian group on the basis $S$. Given a subset $T\subseteq S$, let $H$ be the subgroup of $G(S)$ generated by $T$.
I wonder if the following is true: Can ...
user839372
1
vote
1
answer
120
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What is the free abelian group on $\mathbb{N}$?
I learnt that the free abelian group on a set $X$ is the group $(\operatorname{Hom}(X, \mathbb{Z}), +)$. Okay, this sounds all right, but I also know the famous result that $\mathbb{Z}^{\mathbb{N}}$ ...
- 885
1
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2
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120
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Free abelian group $F_i$ with free basis $B_i$
I have to prove this statement:
Let free abelian group $F_i$ with free basis $B_i$ for $i=1,2$ then $F_1\cong F_2$ iff $\lvert B_1 \rvert = \lvert B_2 \rvert$.
I prove it by the idea that element of ...
- 41
0
votes
1
answer
49
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Exact sequence of 4 abelian groups, 3 of them being free
Let $G$ be a finitely generated abelian group that fits in an exact sequence of the form
$$0 \to \mathbb{Z} \to \mathbb{Z}^n \stackrel{f}{\to} G \stackrel{g}{\to} \mathbb{Z}^m \to 0$$
for some $n \geq ...
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