Is the group isomorphism $\exp(\alpha x)$ from the group $(\mathbb{R},+)$ to $(\mathbb{R}_{>0},\times)$ unique?
116168
Is $e^x$ the only isomorphism between the groups $(\mathbb{R},+)$ and $(\mathbb{R}_{> 0},*)$?
58484
Is the topological product $\Pi \{G_\alpha: \alpha < \kappa\} $ a topological group?
100149
If there are injective homomorphisms between two groups in both directions, are they isomorphic?
158975
Isomorphism of centers of isomorphic groups
116750
On surjectivity of exponential map for Lie groups
88474
$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
133348
Is isomorphism not always unique?
126930
Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$
100439
Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$
142851
Every group homomorphism from $(\mathbb{Q}, +)$ to $(\mathbb{Q}, \times)$ is the trivial map.
99032
Prove that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.
109610
Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$
137949
If the quotient group $G/N$ be isomorphic to $G_2$, is $G$ isomorphic to $G_1\times G_2$ where $N$ is isomorphic to $G_1$?
58707
Prove Quotient Group Isomorphism
127491
Isometric isomorphism between $R^2$ and $R$
99382
What is needed to make Euclidean spaces isomorphic as groups?