Let $G$ and $H$ be abelian groups. Show that the product $G\times H$ is also abelian.
5741
Direct product of abelian groups
154398
Show that H is a subgroup of G
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
91997
Product of two subgroups the whole group?
56245
Let H be a subgroup of a group G. Prove that the following statements are equivalent.
143550
Order of product of two elements in an abelian group
24181
Showing that $G$ is a group under an alternative operation.
57809
For which $n$, $G$ is abelian?
32735
If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
118299
Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
135188
Show that if $\vert G\vert = pq$, then either $G$ is abelian or $Z(G) = \{ e\}$.
74586
Prove that every group of order $4$ is abelian
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
105951
Does $G\times K\cong H\times K$ imply $G\cong H$?
94090
How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$