Elementary proof that centre of finite $p$ group is not trivial
69820
A nontrivial p-group has nontrivial center
113357
Groups of order $pqr$ and their normal subgroups
60208
Proof of property of abelian group
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
62755
The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
63136
Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
66877
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
114320
Showing that if the center of a group is trivial, then the center of its automorphism group is trivial
85222
Direct sum of finitely-many non-abelian groups
110708
On proving that a finitely generated group has a finite number of subgroups with index $n$.
128300
Prove that a group with exactly two proper nontrivial subgroups is isomorphic to $\mathbb{Z}_{pq}$ or $\mathbb{Z}_{p^3}$.
67973
Prove that a finite abelian group is simple if and only if its order is prime.
67554
In a group, the number of non-self-inverse elements is even
54453
Finite abelian $p$-group with only one subgroup size $p$ is cyclic
8746
Groups of order $pq$ without using Sylow theorems
101735
A finite $p$-group cannot be simple unless it has order $p$