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Now lets discuss about the Second chapter named as SUBGROUPS . As mentioned before I am following the sequence of chapters from Herstein.

IMPORTANT IDEAS:

i) First go through the definition very well. You will see that H is a subgroup of G when H is a group under the same operation of G, and H is a subset of G. That’s all.
You only need to remember that H is a subgroup of G iff H is closed under the same operation of G and has an inverse of every element in H.
Now does any question pop out in your mind?……………… If yes then you are on the right track in Group theory but if NO then let me tell you the question,
At the very beginning of this discussion I wrote “H is a subgroup of G when H is a group under the same operation of G, and H is a subset of G.” So where the hell are associative and identity property?
You must be thinking now that hmmmmm huh!!!????

Lets pause and think that if elements of G are associative why wont be H’s elements.
Is the identity same as G’s? Have a look at yourself.

ii) One of the most important ideas of this chapter is understanding the “COSETS”
Cosets are nothing but collection of elements of the form $ha$ or $ah$ where $h\in H, a\in G$