Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
38687
Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
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 $G \cong H \times K$
58205
$ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
78852
If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
104855
Prove that $N \subset Z(G)$
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
81864
Finite group of order $mn$ with $\gcd(m,n) = 1$ .
11957
G is the product of its Sylow subgroups
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.