1
$\begingroup$

I read the answers to these questions:

Proof of Lagrange theorem - Order of a subgroup divides order of the group

Lagrange's theorem

How do I know that the cosets of H have the same number of elements?(That equivalence classes form a partition of G.)

$\endgroup$
2
$\begingroup$

Consider a function defined by $f:H\to Ha$ by $f(x)=xa.$ Then it can be easily verified that $f$ is bijection. i.e., $H$ and $Ha $ has same cardinality.

| cite | improve this answer | |
$\endgroup$
1
$\begingroup$

Hint:

Consider the map $H\to gH$ given by $h\mapsto gh$. Prove that it is bijective.

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

Let $a$ and $b$ be any two elements, $H$ a finite subgroup. Consider $aH$ and $bH$. If $|aH|\neq|bH|$ then there must be two elements in $H$, say $f$ and $g$ such that either $af=ah$ or $bf=bh$. Assume WLOG $af=ah$. Then $f=h$, a contradiction. Thus all the coasts have the same size.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.