Prove that $G$ has a subgroup of order $p^k$ using the third isomorphism theorem and induction on $k$
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
11957
G is the product of its Sylow subgroups
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
61732
Prove any group with 3 elements is isomorphic to $\mathbb{Z}_3$
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
79995
G is group of order pq, pq are primes
61550
Existence of elements of order $p^k$ in a finite group
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
7843
Show that order of $a^k$
38153
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
115663
Proving an isomorphism using the isomorphism theorems
104349
Proof by induction involving symmetric groups
50157
order of a sylow p-subgroup
113357
Groups of order $pqr$ and their normal subgroups