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I've got a problem with the finish of the answer of the following problem: Homology of connected sum of real projective spaces.

How is that $(\mathbb{Z}^{n-1}\oplus\mathbb{Z})/(2,\dots,2)\mathbb{Z}$ with $n$ twos is equal to $\mathbb{Z}^{n-1}\oplus\mathbb{Z}_2$?

$\mathbb{Z}^{n-1}\oplus\mathbb{Z}=\mathbb{Z}^n$,since $n$ is finite and we have finite sequences, right? Or am I wrong?

Thanks in advance!

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1 Answers1

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There is a basis of $\mathbb{Z}^n$ that contains $e=(1,1,\dots,1)$. Using that basis, it is clear that $$\mathbb{Z}^n/(2,\dots,2)\mathbb{Z}=(\mathbb{Z}^{n-1}\oplus\mathbb{Z})/(0^{n-1}\oplus2e\mathbb{Z})\cong \mathbb{Z}^{n-1}\oplus\mathbb{Z}_2$$

lhf
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