What is wrong with the followig proof that $\mathbb{Z}$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$
34758
Cancellation Law for Direct Sums - What is wrong with this argument?
32302
Neukirch’s Number Theory – why is $ℂ \otimes_ℚ K → K_ℂ,~z \otimes a ↦ (j(z)a)_τ$ an isomorphism?
154754
Is $O_n$ isomorphic to $SO_n \times \{\pm I\}$?
26661
Ring homomorphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_m$
128732
$\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups
25050
To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.
107687
How to prove that $U_{2^n}$ is isomorphic as group to $\mathbb Z_2 \times \mathbb Z_{2^{n-2}}$ for $n \ge 3$?
109596
Isomorphism from $\mathrm{Aut}(\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z})$ to $S_3$
67772
How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$
108213
$N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic.
70861
$\mathbb P^n\times \mathbb P^m\not\cong \mathbb P^{m+n}$
92703
Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
128652
Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$
81238
Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?
94747
Show that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic.
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
2235
Showing $\text{Aut}_\mathsf{Grp}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) \cong S_3$