Prologue~
"Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection."
~Hermann Weyl
Verse I~ Groups?
There are two ways one can view a group or for that matter of fact, the very notion of a group.
One, is to think of it as symmetries— a more intuitive and concrete approach, and the other is to start with the abstract axioms and build from there. The latter took a little time for me to undertsand whereas, the former comes intuitively when one makes sense of the latter.
And if I may take the liberty to say, the latter makes you more than prepared for the former.
That is to say that understanding the axioms prepares you well for appreciating the intuitive symmetries.
For somebody looking for a formal definition, a group is a set of objects which follow the closure property, associativity, holds the existence of an identity (unique), as well as an inverse.
Verse II~ Groups...
Now the question arises- what must one exactly think of when talking of a group?
Think of a triangle, or a polygon, or a Rubik's cube, or even a point in space.
While it may have begun to sound confusing as to what a group would be now, but from the former intuition, it is simply a set of symmetries of an object. So all these figures listed above can then be considered as objects and the different rotations or reflections that would make the figures unchanged (or 'invariant') from their initial position (or alignment) , will be the symmetries of the objects, and then these together, form a group. In the same manner, the transformations that preserve the structure of a Rubik's cube or a point in space also constitute groups.
Verse III~ Groups!
Why groups?
In a recent conversation, I stumbled on this sentence that emphasizes the importance of a group- not just to a person in the field of Mathematics, but also Physics, and I would like to paraphrase that here.
'We Physicists when make a statement, we are intuitively talking of groups. Groups are fundamental even in Physics. When we make a statement, it already bases its idea (or logic or concept) on the structure of a group.'
Epilogue~
Now, what do I think of groups—or rather, mathematics in general?
When one begins to understand such abstract notions, they become essential. In fact, they become so fundamental to the understanding of the world around and the world within, that they feel almost quintessential.
And of course, there is art. But, what is mathematics, if not art?