The notation for fundamental group! I am new in algebraic topology and I am wondering why the notation for the fundamental group is $\pi_1$? I mean what is this "1" for?
I searched on the Web and did not find anything!  Any idea?
Also why the name of the fudamental group is fundamental group? 
It is group! ok! But why fundamental group?
Which information it says that it isbdeserved to be called fundamental! 
I know that it has some interesting application between topology and algebra.  But there should be something more interesting. 
 A: This is the first homotopy group in the sense that we are detecting “1-dimensional holes” by examining continuous images of $S^1$ in our space $X$. There are higher homotopy groups $\pi_k$ for all integers $k\ge1$. Actually, for $k\ge 2$ these homotopy groups $\pi_k$ are actually Abelian, but very hard to compute. They are defined analogously, except we look at homotopy classes of maps $S^k\to X$.
As for why it is "fundamental"; the word fundamental basically means foundational and basic. Indeed $\pi_1$ is fundamental in several regards. First, it is the prototype for the higher homotopy groups.
Second, there is a Galois correspondence between covering spaces of $X$ and subgroups of $\pi_1(X,x)$, for $x$ a chosen base point. Basically, good quotients of the universal cover $\widetilde{X}$ by subgroups $H\subseteq \pi_1(X,x)$ give rise to intermediate covering spaces $\widetilde{X}\to Y\to X$ of $X$. So, if one wishes to study Galois correspondences in  the abstract sense (for instance the Étale fundamental group as outlined in SGA) the fundamental group is a good starting point.
Lastly, $\pi_1$ is a "fundamental" invariant of spaces, and allows us to distinguish homotopy types of spaces.
Also, another name for $\pi_1$ is the Poincaré Group for historical reasons.
