When I study topics in group theory (I am currently following Dummit and Foote) I don't care about examples so much. I read them, try to understand the applications of the theorems and corollaries on the examples. Most of the examples are about $D_{2n}, S_n$ and $\mathbb{Z}_n$, etc.
However, if I have difficulty with them, sometimes I skip them (although I mostly try hard to understand them first). The same happens with exercises, I care more about questions which ask me to prove general statements – in the form of corollaries or theorems. For example: "prove that $G$ is abelian if $G/Z(G)$ is cyclic"
So, my question is, what is the importance of examples of group theory? Should I care more about examples? Or will I be okay if I understand the definitions, theorems, corollaries and solve the proof-based exercises well?
Also, is there a reference – a book or a collection of sites – which talks about the groups only?
I mean it talks about, for example, $D_{2n}$, Matrix groups , $S_n$ , Klein four-group etc and the relations between those groups, their properties, how they act on quotients, the relation between the center and factor of the group and their subgroups, their normal subgroups, their sylow p-subgroups and so on.
If no such reference exists, then I have to collect the information of each group from the sections and exercises and try to summarize these information of each group which is mentioned in the text.
Best Answer
There is some pedagogical value to understanding how theorems/definitions apply to and playout with particular groups (i.e., through examples). While it is admirable to yearn for a thorough theoretical understanding of group theory, try to think of studying and probing the examples as a means of testing your understanding of the theories and definitions, as they relate to specific groups. Being able to construct a proof is wonderful, but being able to use and apply your learning is crucial to obtaining a fluency with what you're learning. Being able to understand and apply the theorems and definitions is also a test of your comprehension, and will facilitate the proof-writing process as well. For example, being able to construct and anticipate counterexamples is crucial to proof-writing, and counterexamples will be hard to come by if you don't understand how particular groups exemplify or fail to exemplify group properties.
A great website is the website Groupprops - "A Group Properties Wiki". You'll find a menu to the left:
and can quickly look-up (almost) anything/everything you'd want to know about (almost) any group.