Let $G$ be a finite group of even order. Prove that there exists $a \in G$ of order $2$.
26398
Group of even order contains an element of order 2
121423
Prove that every element of a finite group has an order
67554
In a group, the number of non-self-inverse elements is even
8742
Examples and further results about the order of the product of two elements in a group
101735
A finite $p$-group cannot be simple unless it has order $p$
76943
Existence of subgroup of order six in $A_4$
134676
Number of elements of order $p$ is a multiple of $p-1$ (finite group).
17178
Order of $\phi(g)$ divides the order of $g$
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
104080
Group with order $pq$ has subgroups of order $p$ and $q$
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
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 $P_i\leq G_i$?
126893
A group of order 2p (p prime) and other conditions - prove abelian.
112974
The $h-$th power of every element in a finite group of order $h$ is the identity element of the group
15526
Question on groups of order $pq$