All Questions
Tagged with free-groups free-abelian-group
38
questions
2
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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, ...
1
vote
0
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55
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Construction of Free abelian groups on Massey Book
I am reading the book of Massey of Algebraic topology, and I am having trouble to understand this construction.
Let $ S = \left\{ x_i : i\in I \right\}$. For each index $i$, let $S_i$ denote the ...
- 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
0
votes
3
answers
281
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Abelianization of free groups
I'm reading Hatcher's Algebraic Topology and I have some questions about an argument on Page 42:
The abelianization of a free group is a free abelian group with basis the same set of generators, so ...
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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
0
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0
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57
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Order of a quotient of a free abelian group
Let $G\subseteq \mathbb{C}$ be a free abelian group of rank $n$ and let $p$ be a prime. Then, we know that $$|G/pG|=p^n$$ and in fact for this we don't even need $p$ to be a prime. Suppose now that ...
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2
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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
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0
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142
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In $F=F^{ab}(A)$, define $f\sim f'$ if and only if $f-f'=2g$ for some $g\in F$. Show $F/\sim$ is finite if and only if $A$ is finite.
this is a question from Aluffi's Algebra: Chapter 0, Exercise II.5.10. The full question is the following:
Given a free abelian group over a set $A$, denoted $F=F^{ab}(A)$, we define an equivalence ...
- 51
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
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2
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64
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Basis for a free abelian group $F(X)/F(X)'$
Let $|X|=n,$ prove that $F(X)/F(x)'$ is a free abelian group of rank $|X|$. For the rank prove and use that $\{xF(X)' : x \in X \}$ is a basis for $F(X)/F(X)'$.
The abelian part is because in a group $...
- 549
2
votes
1
answer
126
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Rank of a free group times a free abelian group.
I know that the rank (i.e. minimal number of generators) of the product $\mathbb{Z}\times F_2$, of the infinite cyclic and free group on two generators, is three, but the only argument I could quickly ...
- 243
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
2
votes
4
answers
388
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How's the 'integer lattice' or the 'direct sum' of $\mathbb{Z} \oplus \mathbb{Z}$ not a Free group?
My question is about the direct sum $\mathbb{Z} \oplus \mathbb{Z}$ which is a Free Abelian group and not a free group.
The the integer lattice, or what I think is the direct sum $\mathbb{Z} \oplus \...
- 69
0
votes
1
answer
108
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Factor group of a free group
Let $F[A]$ be the free group on the generating set $A$. Let $C$ be the commutator subgroup of $F[A]$, then show that $F[A]/C$ is a free abelian group with basis $\{aC \mid a \in A\}$. It is trivial ...
- 711
2
votes
1
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307
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Free groups as free product of infinite cyclic groups
Let $S$ be an arbitrary set (countable or uncountable). It is clear that the free abelian group generated by $S$ is isomorphic to the direct sum
$$\bigoplus_{s\in S}\mathbb{Z}.$$
Is the free group ...
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