$G$ is a group, with $|G| \in \mathbb N, a \in G,a^{|G|} = 1$. True for non-abelian groups?
63769
Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
41977
If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
71035
Non abelian subgroup of a abelian group.
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
80849
There does not exist group $G$ such that Aut$(G)\cong \mathbb{Z}_n$ (for odd $n$)
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
112157
If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
56650
Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
57809
For which $n$, $G$ is abelian?
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 $ab\in H$.
108309
An example of a non-abelian group of order $p^3$
6153
Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian.
4838
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
107511
If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$?