Group-theoretic proof that $n$ does not divide $2^n-1$.
4970
For $n \geq 2$, show that $n \nmid 2^{n}-1$
90329
Prove there is no simple group of order $729$
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
130093
Prove by Induction : $n < 2^n$
112191
How can I show $q^k$ does not divide ${{n}\choose{q}}$ when $q$ is a prime factor of $n$ and $k$ is the largest integer such that $q^k$ divides $n$?
86542
Group of positive rationals under multiplication not isomorphic to group of rationals
151692
When does $2^n-1$ divide $3^n-1$?
46904
Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group.
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
114873
Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.
89989
A set of integers whose elements all divide $2015^{200}$ but do not divide each other
131385
How to provide a counterexample for if $n$ is prime, then $2^n -1$ is prime
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
100305
Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff...
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
77076
Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
95764
Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.