Prove that for a group with even order $2k$, $k$ odd, there is a subgroup $K$ with order $k$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
53409
Show group of order $4n + 2$ has a subgroup of index 2.
51747
Showing that the intersection of all subgroups of order $n$ is normal
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 $d$.
16476
Order of the product of two commuting elements with coprime orders in a group.
126113
Prove the existence of order 4 subgroups of order 8 groups
2174
Proving that HK is a subgroup when K is normal
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
12667
How many subgroups of order $n$ does $D_n$ have?
41231
Subgroups of (Z,+) of order n
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
2794
Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.
37971
Prove that the center of a group is a normal subgroup
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
110708
On proving that a finitely generated group has a finite number of subgroups with index $n$.
152630
showing that a group of order 45 is abelian