Prove the following statement: "Let $G$ be a group, and $N ⊲ G$. Then $G/N$ forms a group under the operation $(gN)(hN) = ghN$ "
112407
Set of left cosets is a group
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
51896
Prove that: set $\{1, 2, 3, ..., n - 1\}$ is group under multiplication modulo $n$?
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 $G \cong H \times K$
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
158490
Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group of G
112157
If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
104855
Prove that $N \subset Z(G)$
90332
Prove that that $U(n)$ is an abelian group.
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
24181
Showing that $G$ is a group under an alternative operation.
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
46260
Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
140471
Is $ G \cong G/N \times N$?