For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
26398
Group of even order contains an element of order 2
108330
Product of all elements in finite abelian group equals its own inverse
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
61550
Existence of elements of order $p^k$ in a finite group
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
13355
Product of elements of a finite abelian group
36887
Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
86626
Finite fields, existence of field of order $p^n$,proof help
46260
Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
40846
When can a (finite) group be written as the quotient of some other group by its center?
158282
Finite order of an element $f(g)$ of a homomorphism $f$
36882
Two finite abelian groups with the same number of elements of any order are isomorphic
43059
Show that $g^k=e$ if and only if the order of $g$ is finite and divides $k$.
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
114590
Prove that any finite group $G$ of even order contains an element of order $2.$
1577
Order of nontrivial elements is 2 implies Abelian group
56234
Isomorphism $f$ preserves the order of an element?