$G$ a finite $p-$group, $H < G$ (i.e, $H$ a subgroup of $G$, but $H \neq G$), prove that $H \neq N_G(H)$
103423
Normal subgroup where G has order prime
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
146760
Prove that H is a subgroup of G
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.
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
112974
The $h-$th power of every element in a finite group of order $h$ is the identity element of the group
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
39469
If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
113357
Groups of order $pqr$ and their normal subgroups
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
61550
Existence of elements of order $p^k$ in a finite group
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$