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This is question related to following link:

Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

$M$ and $N$ as in the linked post, that is, $M=\mathbb{Z}$ an $N=\bigoplus_{\mathbb{N}}\mathbb{Z}/2\mathbb{Z}$.

So you constructed an ses $S_1: 0\longrightarrow M\stackrel{\alpha}\longrightarrow M\oplus N\stackrel{\beta}\longrightarrow M\longrightarrow 0$ that is not split. Thus $S_1$ cannot be equivalent with $S_0: 0\longrightarrow M\stackrel{i}\longrightarrow M\oplus N\stackrel{\pi}\longrightarrow M\longrightarrow 0$ in which $i$ and $\pi$ are the natural injection and natural projection, respectively. Therefore, there cannot be an isomorphism $\phi : M\oplus N \longrightarrow M\oplus N$ such that $\phi \circ \alpha = i$.

Is it possible to prove that the identity is the only isomorphism $M\oplus N \longrightarrow M\oplus N$ ? I will be grateful if you / someone can prove that for me.

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Steenis
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    Any non-trivial automorphism of $M$ and/or $N$ will give rise to a non-trivial automorphism of $M\oplus N$, for example. And of course, in case $M=N\ne1$ we may swap the summands as non-trvial automorphism – Hagen von Eitzen Aug 08 '16 at 13:51
  • Yes, M and N as in the link, sorry this is very unclear. My Original post was more clear, but the moderators wanted me to move the post. – Steenis Aug 08 '16 at 16:12
  • @Hagen: sorry. M and N as in tyhe linked post, I am sorry that that is confusing. I thought that there is only one automorphism of $\Bbb Z$ and only one of $\Bbb Z_2$, so is there only one automorphism of $M\oplus N$ – Steenis Aug 08 '16 at 16:18

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Your question is whether the $\mathbb{Z}$-module $\mathbb{Z}\oplus \bigoplus_{\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$ admits automorphisms other that the identity.

As Hagen von Eitzen says in a comment, any non-trivial automorphism of $\mathbb{Z}$ or of $\bigoplus_{\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$ will give rise to such an automorphism. Since $\mathbb{Z}$ has a non-trivial automorphism (given by multiplication by $-1$), this answers your question.

Note that $\bigoplus_{\mathbb{N}} \mathbb{Z}/2\mathbb{Z}$ has even more non-trivial automorphisms.

Pierre-Guy Plamondon
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