If $p$ is an odd prime then prove that there cannot exist a finite group $G$ such that ${\rm Aut}(G)\cong \mathbb{Z}_p$.
135710
Cyclic Automorphism group
140294
prove that $n >2$ and $n$ is a prime then $n$ is odd
66877
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
151786
Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$
38153
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
150007
Prove that $(\mathbb{Z}_n , +)$, the integers $\pmod{n}$ under addition, is a group.
89382
In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
92057
Prove that for every odd integer n, there is a group...
145254
$G$ be a non-nilpotent and supersoluble finite group.
42932
Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
126494
Prove that there doesn't exist any integer $x \ge 3$ such that $x^2-1$ is prime.
38805
Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
114590
Prove that any finite group $G$ of even order contains an element of order $2.$
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
154305
Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
122335
Abelian $p$-group with unique subgroup of index $p$