Use Lagrange's theorem to prove that if $G$ is a finite group of order $n$ then the map $x \to x^k$ is surjective.
75473
If a group $G$ has odd order, then the square function is injective.
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
104855
Prove that $N \subset Z(G)$
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
152662
Proving that any finite cyclic group of order $n$ is isomorphic to $$
110708
On proving that a finitely generated group has a finite number of subgroups with index $n$.
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
49953
Proving that a function from $N\times N$ to $N$ is bijective.
108691
Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
64269
Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
65744
How can I Prove that H=G or K=G
46116
Proof of the First Isomorphism Theorem for Groups
67234
The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
81864
Finite group of order $mn$ with $\gcd(m,n) = 1$ .