Sufficient conditions for $G\cong N\times G/N$
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
48393
Abelian groups of order n.
137739
Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
120642
Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
98765
Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$
40094
Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?
128652
Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$
75068
subgroups of $\mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z$
151130
When is the product of $n$ subgroups a subgroup?
77128
Necessary and Sufficient conditions for $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
119793
Proving that $D_{12}\cong S_3 \times C_2$