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I've been reading group theory from Rotman's book "Introduction to the theory of groups" and in Chapter 10 of free abelian groups there is an exercise which am having a hard time to prove.

The exercise is the following:

An Abelian group $F$ is free if and only if it has the projective property.

Necessary definitions:

Free Abelian: An Abelian group $F$ is free abelian if it is the direct sum of infinite cyclic groups.More precisely, there is a subset $X\subset F$ of elements of infinite order called a basis of $F$ with $F=\bigoplus\limits_{x \in X}\langle x\rangle.$

Projective Property: We say that an Abelian group F has the projective property if for every two abelian groups $B,C$ and for every $\beta:B\to C$ surjective homomorphism and every $\alpha:F\to C$ homomorphism there exists homomorphism $\gamma:F\to B$ such that $\alpha = \beta \circ \gamma.$

Rotman has the proof for the $"\implies"$ direction and its understandable although for opposite direction i dont know how to start, i picked $B,C$ to be $\bigoplus\limits_{x \in F}\langle x \rangle$ and $\beta(x)=(\beta_y)_{y \in F}$ where $\beta_y = x$ if $y=x$ and $\beta_y =0$ for $y\neq x$, $\alpha(x) =(\alpha_x)_{x \in F}$ where $\alpha_x =x$ and $\alpha_y = 0 $ for $y\neq x$ and then $\gamma:F \to B$ seems to be an injection hence $F \cong \operatorname{Im}\gamma \leq\bigoplus\limits_{x \in F}\langle x \rangle$ ... is this correct? If not do you have any ideas on how am going to prove it?

Thanks in advance !

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  • $\begingroup$ Use $\langle X\rangle$ for $\langle X\rangle$. $\endgroup$
    – Shaun
    Feb 2, 2019 at 11:03
  • $\begingroup$ Do you know about presentations? $\endgroup$
    – Shaun
    Feb 2, 2019 at 11:04
  • $\begingroup$ Use $\operatorname{Im}$ for $\operatorname{Im}$. $\endgroup$
    – Shaun
    Feb 2, 2019 at 11:05
  • $\begingroup$ A quick glance at the content section of the fourth edition of the book gives me the sense that presentations aren't covered before nor during its Chapter 10. Besides, the result in question seems like it would be used to help understand presentations during an introduction to them. $\endgroup$
    – Shaun
    Feb 2, 2019 at 11:10
  • $\begingroup$ Ah, I see; Chapter 11 introduces presentations. $\endgroup$
    – Shaun
    Feb 2, 2019 at 11:11

1 Answer 1

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Let $A$ be a projective Abelian group, consider the free Abelian group $F_A$ whose basis is $\{[x],x\in A\}$, there exists a canonical projection $p:F_A\rightarrow A$ defined by $p([x])=x$. Since $A$ is projective, there exists $f:A\rightarrow F_A$ such that $p\circ f=Id_A$, we deduce that $f$ is injective and $A$ is isomorphic to a subgroup of $F_A$. A subgroup of a free Abelian group is free.

https://en.wikipedia.org/wiki/Free_abelian_group#Subgroups

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  • $\begingroup$ $F_A$ is free abelian or just free? $\endgroup$ Feb 2, 2019 at 16:02
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    $\begingroup$ it is free Abelian $\endgroup$ Feb 2, 2019 at 16:07
  • $\begingroup$ Ok i see, so i guess your $F_A$ is mine $\bigoplus\limits_{x \in A}\langle x \rangle$ $\endgroup$ Feb 2, 2019 at 16:11

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