Let $G$ be a finite abelian group. Show that there is a Galois extension $K/\Bbb Q$ with $\text{Gal}(K/\Bbb Q) \cong G$.
42153
Every finite abelian group is the Galois group of some finite extension of the rationals
135611
Extensions of $\mathbb{Z}$ by $\mathbb{Z}$
76317
Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
105103
Automorphism from $\Bbb Q$ to $\Bbb Q$
4996
Proving $Gal(K/\mathbb{Q})$ is $D_4$.
27905
Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
100439
Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$
127330
Exercises on Galois Theory
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
13631
Showing group with $p^2$ elements is Abelian
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
41815
Galois extension of intersection of fields
51352
Subgroups of a finite abelian group
81354
The product of all the elements of a finite abelian group
117789
Computing Galois Group of $\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}$
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.