Showing a Group $G$ is not Simple
12066
Group of order pq is not simple
39425
Surprising but simple group theory result on conjugacy classes
51196
A group of order $p^2q^2$ is never simple
85018
When is the automorphism group of the Cayley graph of $G$ just $G$?
129830
A group with no proper non-trivial subgroups
67554
In a group, the number of non-self-inverse elements is even
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
61373
A normal subgroup that is not a characteristic
102912
Subgroups of finitely generated groups are not necessarily finitely generated
65613
$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.
11957
G is the product of its Sylow subgroups
33912
Group with 380 elements is not simple.
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
90616
When is mapping $g$ to $g^{-1}$ a group homomorphism?
56650
Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
103216
The set $\{g^2 | g \in G\}$ in a group $G$