If $a, b$ are in group and $ab$ has finite order $n$, why does $ba$ have order $n$ as well?
34973
Is it true that the order of $ab$ is always equal to the order of $ba$?
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
933
How many groups exist with order $n$ (two isomorphic groups are treated as the same group)
1577
Order of nontrivial elements is 2 implies Abelian group
12667
How many subgroups of order $n$ does $D_n$ have?
150156
Proving that $S_n$ has order $n!$
17178
Order of $\phi(g)$ divides the order of $g$
145406
Prove every element of $G$ has finite order.
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
26398
Group of even order contains an element of order 2
76200
Is the rank of $AB$ always equal to the rank of $BA$?
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
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 $n$
8742
Examples and further results about the order of the product of two elements in a group
105921
Why do we have $ab=ba$?
60075
Does there exist an $n$ such that all groups of order $n$ are Abelian?
122959
How is number of conjugacy class related to the order of a group?