# Properties of Cosets

Theorem 1: If $h \in H$, then the right (or left) coset $Hh$ or $hH$ of $H$ is identical to $H$, and conversely.

Proof: Let $h’$ be an arbitrary element of $H$ so that $hh’ \in hH$.

Again, since $H$ is a subgroup, we have
$h \in H,\,\,\,h’ \in H\, \Rightarrow \,hh’ \in H$

Thus every element of $hH$ is also an element of $H$.

Hence $hH \subset H\, — \left( i \right)$

Again $h’ = \left( {h{h^{ – 1}}} \right)h’ = h\left( {{h^{ – 1}}h’} \right) \in hH$

This shows that every element of $H$ is also an element of $hH$.

Hence $H \subset hH\,\, — \left( {ii} \right)$

From (i) and (ii) it follows that $hH = H$

Similarly, we can show that $Hh = H$

Conversely, $Hh = H \Rightarrow eH \in H \Rightarrow h \in H$ and similarly $hH \Rightarrow H \Rightarrow h \in H$.

Theorem 2: Any two right (or left) cosets of $H$ are either disjoint or identical.

Proof: Let $H$ be a subgroup of a group $G$ and let $aH$ and $bH$ be two left cosets. Suppose these cosets are not disjoint. Then they possess an element, say $c$, in common. Then $c$ may be written as $c = ah$, and also as $c = ah’$, where $h$ and $h’$ are in $H$.

Therefore,
$ah = bh’$
$a = bh'{h^{ – 1}}$

Since $H$ is a subgroup, $h'{h^{ – 1}} \in H$.

Let $h'{h^{ – 1}} = h”$ then $a = bh”$. Hence $aH = \left( {bh”} \right)H \Rightarrow aH = b\left( {h”H} \right) = bH$

Therefore the two left cosets are identical if they are not disjoint.

Thus either $aH \cap bH = \phi$ or $H = bH$

A similar result can be shown to hold for right cosets.

Theorem 3: If $H$ is finite, the number of elements in a right (or left) coset of $H$ is equal to the order of $H$.

Proof: The mapping $f:H \to Ha$, defined by $f\left( {{h_i}} \right) = {h_i}a$, is obviously onto.

It is one-one, since
$f\left( {{h_i}} \right) = f\left( {{h_j}} \right) \Rightarrow {h_i}a = {h_j}a$
$\Rightarrow {h_i} = {h_j}$

From the right cancellation law it is onto, since an element $ha$ belonging to $Ha$ is the $f –$ image of $h$ belonging to $H$.

It follows that the number of elements in a left coset of $H$ is the same as that of $H$.
Similarly the number of elements in  a left coset of $H$ is the same as that of $H$.