If $G$ is a group and $H<G$ and $|G:H|=2$ prove that for each $a\in H$, $aH=Ha$
58914
Any group of index 2 is normal
9110
Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
11003
Proving that $G/N$ is an abelian group
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
147253
Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
154398
Show that H is a subgroup of G
77361
How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
101345
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
71420
Prove that a group is abelian.