If $a,b \in G$ and has finite prime order, what is the order of ab?
74187
$\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
28676
Is a finite group uniquely determined by the orders of its elements?
83721
If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
3703
Order of finite fields is $p^n$
22248
$G$ solvable $\implies$ composition factors of $G$ are of prime order.
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
8434
Order of $\mathbb{Z}[i]/(1+i)$
81116
$gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
50157
order of a sylow p-subgroup
38853
Cyclic group of prime order
118849
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
37913
Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
121481
Non-abelian finite $p$-group in which every element has prime order?
834
If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.
158799
Order of $G$ is 35, are there elements of order 5 and 7 in G?
143550
Order of product of two elements in an abelian group