Finite group $(G, \cdot)$ and H a subgroup of G. Prove that if $x \notin H, y \notin H$ then $xy \in H$.
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
93666
Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
124709
Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G?
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 $G \cong H \times K$
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
45310
If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
13742
Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
119512
If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.