Let $G$ a group and $g,h∈G$. Prove that $|gh|=|hg|$.

Question

A hint would be appreciated.

score 10 · Accepted Answer · answered Mar 16 '13 at 16:41

10

Hint. $$g\color{red}{hghghghghghghghghg}h$$

answered Mar 16 '13 at 16:41

Alexander Gruber

score 4 · Answer 2 · answered Mar 16 '13 at 16:38

4

Hint: show that $hg$ and $gh$ are conjugated.

answered Mar 16 '13 at 16:38

Myself

(Two elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $gag^{-1}=b$.) – user1729 Mar 16 '13 at 16:41

Abhra Abir Kundu · Answer 3 · 2013-03-16T16:49:20.047

0

If $|gh|=k\in N,k\ne 1$(for $k=1$ is trivial) then,

$e=(gh)^k=g(hg)^{k-1}h$

$\Rightarrow (hg)^{-1}=g^{-1}h^{-1}=(hg)^{k-1}$(by multiplying with $g^{-1}$ and $h^{-1}$ on the left and right of th above eqn.

$(hg)^k=e$(Multiplying with $hg$ on both sides of th above eqn.)

Similarly for the other case by reversing the roles for $h$ and $g$.

edited Mar 16 '13 at 16:49

answered Mar 16 '13 at 16:44

Abhra Abir Kundu

3 Answers3