Group of order $2m$ where $m$ odd has a subgroup of index 2.
53409
Show group of order $4n + 2$ has a subgroup of index 2.
28210
A finite group with exactly $2$ conjugacy classes isomorphic to $\mathbb{Z}_2$
60982
Group where every element is order 2
155502
Index of a maximal subgroup in a finite group
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
113936
Showing that a finite abelian group has a subgroup of order m for each divisor m of n
94999
A subgroup with an index of 2 is normal to the group.
2731
Groups of even order have an element whose order is 2.
98474
Show the existence of a certain subgroup of F2
94420
Dihedral group - elements of order $2$
102971
Product of all elements of a finite group with an unique element of order 2
52089
Group of $2n$ elements, $n$ odd, is not simple
156319
Show that a group with $|G| = 33$ contains an element of order $3$
127996
$M_1$ and $M_2$ are subgroups and $M_1/N=M_2/N$. Is $M_1\cong M_2$?
103423
Normal subgroup where G has order prime
58914
Any group of index 2 is normal
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.