I.N. Herstein in Page 34 (last line) and Page 35 of "Topics in Algebra" book goes on to explain a definition of right coset and a lemma like this:
Def: If $H$ is a subgroup of G, and $a \in G$, then $Ha = \left \{ha|h\in H \right \}$;then $Ha$ is the right coset of $H$ in $G$
Lemma: FOr all $a \in G $ $Ha = \left \{x \in G |a \equiv x mod H \right \}$
He goes on to define a set $[a]$ exactly like $Ha$ and trying to show $Ha \subseteq [a] $
My confusion:
Whats going on here?
More specifically, what the lemma trying to convey and why did the author go on to define $[a]$ exactly like $Ha$ and trying to show $Ha \subseteq [a] $ Isnt it trivial that every set is a subset of itself?
If you have the proof of the lemma with you, can you help me understand it. I am not able to understand why exactly are we dealing with $a(ha)^{-1}$ which I understand as motivated from $a = ha mod H$
Thanks for your time and patience Soham