Given a finite group $G$ of order an even number, prove that $G$ contains a non-identity element of order $2$.
26398
Group of even order contains an element of order 2
108830
Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
53409
Show group of order $4n + 2$ has a subgroup of index 2.
58397
Every cyclic group with order > 2 has at least 2 distinct generators
8742
Examples and further results about the order of the product of two elements in a group
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
134376
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
134676
Number of elements of order $p$ is a multiple of $p-1$ (finite group).
5226
Prove that a finite semigroup G with identity is a group if and only if G contains only one idempotent
121077
Product of all elements in finite group
43058
If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)