If $H \leq G$ such that $[G:H]=2$ then $a^2 \in H $ for every $a\in G $
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
54369
Proof of $n^2 \leq 2^n$.
40094
Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
42355
Double check $G\sim H$ iff $G≈H$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
45310
If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
153532
Proving $a^2$ is even $\implies$ $a$ is even by contradiction
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.