TITLE: What is the importance of examples in the study of group theory?
QUESTION [17 upvotes]: 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.

REPLY [3 votes]: You wrote: 
"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"
In any field of research mathematics, it is very rare that what you do is to follow someone elses instructions "prove this". Instead, what you do is to explore the field to try to discover true facts, and once you have convinced yourself of the truth of those facts you then try to prove them. The discovery process is, I think, literally impossible without knowing lots of examples, looking amongst them for patterns, and learning how to construct new examples and explore them too.
Also, from what you write it sounds like you prefer statements that have the logical structure "Prove that for all groups that satisfy such-and-such, they also satisfy so-and-so". Trouble could occur if you were to run into such a statement which was false. You would then be faced with proving a statement having a different logical structure: "Construct a group that satisfies such-and-such and does not satisfy so-and-so". 
It would be more instructive to seek out exercises which are structured somewhat differently: "Is it true that all groups satisfying such-and-such also satisfy so-and-so? If so, give a proof. If not, give a counterexample." Questions with this structure portray research mathematics more realistically. But without having a hearty collection of examples in your pocket ready to use, you might have a hard time handling the "If not..." part of such questions.