From my teacher's notes:
"Suppose $H$ is a subgroup of $G$. For any $a\in G$, we define its associated left coset and right coset to be $$aH = \{ab | b \in H\}$$ $$Ha = \{ba | b\in H\}$$ respectively."
What is a coset? What are the differences between left and right?
Also from the notes:
"Suppose $H$ is a subgroup of $G$. The left cosets $\{aH | a\in G\}$ define a partition of $G$, so are the right cosets."
What does "define a partition of $G$" mean?
Best Answer
Let $G$ be a group and $H$ be a subgroup of $G$. Let $a\in G$. The left coset of $H$ in $G$ with respect to $a$ is the set $$aH=\{ah:h\in H\}$$ while the right coset of $H$ in $G$ with respect to $a$ is the set $$Ha=\{ha:h\in H\}$$
For $a,b\in G$, $ab=ba$ is not necessarily true, that is, $G$ is not necessarily abelian, so $Ha$ and $aH$ are different set.
For example, take $G=S_3$ and $H=\{1,(12)\}$. Try to verify that $H(13)\neq (13)H$.
Let $\mathcal{A}=\{S_i:i\in I\}$ be a family of non-empty subsets of $G$. $\mathcal{A}$ partitions $G$ if
(i)For $i\neq j$, $S_i\cap S_j=\phi$
(ii) $\bigcup_{i\in I}S_i=G$
In other words, every element of $G$ belongs to exactly one set $S_i$ for some $i\in I$. So in this case, $\mathcal{A}=\{Ha:a\in G\}$ fulfils the conditions and form a partition of $G$.