$G$ is a group and $N,M$ are normal subgroups of $G$. Prove that $nm=mn$ for all $n\in N,m\in M$.
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
51359
Prove that every subgroup of an abelian group is a normal subgroup.
80273
Prove the centralizer of an element in group $G$ is a subgroup of $G$
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
7151
Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
81864
Finite group of order $mn$ with $\gcd(m,n) = 1$ .
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
137739
Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
38687
Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
32735
If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
90437
Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
122266
equivalent definition of normal subgroup
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.