Prove that if $G$ is a finite group, then the index of $Z(G)$ cannot be prime.
7528
Index of the center of a group in the group is not a prime number
80849
There does not exist group $G$ such that Aut$(G)\cong \mathbb{Z}_n$ (for odd $n$)
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
102622
K is a subgroup of G. Show that K is not a normal subgroup
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
142228
Proof that a subset closed under group operation of a finite group is a subgroup
48273
If a maximal subgroup is normal, it has prime index
36147
A finite group is nilpotent iff two elements with relatively prime order commute
67973
Prove that a finite abelian group is simple if and only if its order is prime.
121423
Prove that every element of a finite group has an order
38853
Cyclic group of prime order
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
18182
An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
62635
Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$.
79995
G is group of order pq, pq are primes