Show that $HK\cong H\times K$
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
118299
Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$
52105
Prove that $HK$ is a subgroup iff $HK=KH$.
71245
$2 \times 2 $ Orthogonal Group
52517
Construction of $\mathbb{CP}^2$ from $S^2\times S^2$
51898
Show that $\operatorname{Hom}_{R} (R,M) \cong M$
94469
How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
151621
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
139595
Show that $(l_1)^* \cong l_{\infty}$
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
95686
$Lie(G \times H)\cong Lie(G)\oplus Lie(H)$
35948
How to prove $D^n/S^{n-1}\cong S^n$?
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
73235
Rings $R/I \cong h(R)/h(I)\,$ for injective ring hom $h$
130081
Show that $A_n$ is the kernel of a group homomorphism of $S_n \rightarrow \{−1,1\}$.
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
98352
If $k|n$, then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$