prove that the finite group G has a fixed point in S if |G|=$p^r$, p prime, |S|=N, (p,N)=1
13153
at least one element fixed by all the group
66877
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
128305
$PGL(n, F)=PSL(n, F)$
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
134676
Number of elements of order $p$ is a multiple of $p-1$ (finite group).
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
117744
$p$-Sylow in quotient groups
70001
Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
13742
Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
156319
Show that a group with $|G| = 33$ contains an element of order $3$
37331
Prove that every set with more than one element has a permutation without fixed points
135465
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
108830
Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
147253
Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
121077
Product of all elements in finite group