$G$ is a finite abelian group. $|G| = p^n$ where $p$ is a prime. Show that $G$ has a subgroup of order $p^r, r \leq n$ without Sylow Theorems.
113936
Showing that a finite abelian group has a subgroup of order m for each divisor m of n
133770
Subgroups of finite abelian groups.
11957
G is the product of its Sylow subgroups
7973
Groups of order $pq$ have a proper normal subgroup
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
157841
Existence of a non-abelian group of order $p^n$.
79995
G is group of order pq, pq are primes
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
13631
Showing group with $p^2$ elements is Abelian
138108
If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.
125640
Group of order $p^2$ is abelian.
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
70001
Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
66877
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
63499
Groups of prime squared order