$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
60982
Group where every element is order 2
145254
$G$ be a non-nilpotent and supersoluble finite group.
28676
Is a finite group uniquely determined by the orders of its elements?
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
117468
$o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
47478
Show that any cyclic group of even order has exactly one element of order $2$
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
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 $D_p$
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
156319
Show that a group with $|G| = 33$ contains an element of order $3$
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple