Suppose $p$ is prime and $G$ is a group and $|G| = p(p + 1)$. Prove $G$ has normal subgroup of order $p$ or a normal subgroup of order $p + 1$.
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
7108
Normal subgroup of prime index
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
100605
let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
134376
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
126893
A group of order 2p (p prime) and other conditions - prove abelian.
152481
If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
7973
Groups of order $pq$ have a proper normal subgroup
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
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 $D_p$
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
39976
If a group G has only the trivial group and G itself as its subgroup, can we be sure that G has prime order?