# Properties of Cosets

Theorem 1: If $h \in H$, then the right (or left) cosets $Hh$ or $hH$ of $H$ is identical with $H$, 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

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

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,

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 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

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

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