Can a group be thought of as some kind of "rule book" telling how symmetries interact with each other? I'm new to Abstract Algebra and I'm trying to explain myself what a group really is. So far I've looked into a couple of books, read a little about groups here and there and this is what I've understood so far.

Symmetry: A symmetry of an object $O$ is a bijective function $s:O \to O$ i.e. $s(O)=O$

Intuitively, they are transformations that leave the shape of the object unchanged, they just shuffle around the pieces of the object.
Here, I've taken the object $O$ to be the set $O$ since it is my understanding that everything is a set.

Group: A group $G$ is collection of symmetries of the object $O$ satisfying:

*

*It has a symmetry which does nothing to the object $O$

*Every symmetry applied to $O$ can be reversed.

*Any sequence of symmetries applied to $O$ is also a symmetry of $O$ present in $G$

*Given a sequence of symmetries i.e. $s_1,s_2, s_3$, the order in which you apply this sequence of symmetries to $O$ doesn't matter i.e.
$(s_1 \cdot s_2) \cdot s_3 = s_1 \cdot (s_2 \cdot s_3)$

Hence a Group is merely a "map" or some kind of "guide book" that tells you "how" the symmetries interact with each other and you can just throw away the object, you don't care about it.
Is my intuition so far correct? If there are any flaws in it, please point out. If you can improve on this intuition, please by all means.

Now my questions are,

*

*Why do you not care about the object?

*Given a group, how can I find the object that this group describes the symmetries of?

*Does every group describe symmetries of some object?


 A: Your intuition is good, but there are some problems. I must admit that I have no idea about the meaning of “Any sequence of symmetries applied to $O$ is also a symmetry of $O$ present in $G$”. Perhaps that you meant that the composition of symmetries present in $G$ also belongs to $G$.
Note that every group $G$ is a set of symmetries of $G$ itself: for every $h\in G$, consider the map$$\begin{array}{ccc}G&\longrightarrow&G\\g&\mapsto&hg.\end{array}$$The set of all these maps is a set of symmetries of $G$. So, yes, in this sense every group is the group of symmetries of some object.
However, thinking this way will perhaps make harder to see that quite often the same group is a group of symmetries of two very distinct objects. For instance, the group of all isometries of a regular triangle and the group of all permutations of the set $\{1,2,3\}$ are the same group (technically speaking, they are isomorphic), although, of course, we have very distinct objects here.
A: 
Is my intuition so far correct?

Pretty much, yeah!

Why do you not care about the object?
Two reasons:


*

*Because groups are, by themselves, interesting objects.

*Because by not caring about the particular object, everything you discover about its group of symmetries holds for every object with the same symmetry group!


Given a group, how can I find the object that this group describes the symmetries of?
Does every group describe symmetries of some object?

It's... Complicated.
