Intuition behind Upper central series of Group To construct abelian groups from non-abelian groups people have introduced the notion of Commutator subgroup and solvability of groups.
Can somebody explain what is the intuition behind in introducing Upper central series and central series. 
 A: Subnormal series are basically ways of breaking up a group into pieces. How the pieces are put together exactly is a complicated story, though. It would probably be more accurate to only use the term "piece" to describe subgroups, and quotients of a group can instead be thought of as collapsed versions of the original group. A more scenic metaphor is the following: smaller subgroups can be thought of as zooming in closer to the identity of the group with a camera, and taking quotients amounts to "refocusing" the camera $-$ the smaller the group is collapsed (equivalently, the bigger the normal subgroup being divided by is), the lower resolution a picture we have of the group. Thus a subnormal series provides a way of seeing the group at different scales and resolution.
There are two dual ways of measuring the deviation from abelianness $-$ commutators and centrality. In the first case, $[a,b]=e$ iff $a,b$ commute, so the derived subgroup $G'=[G,G]$ is generated by the "errors," the fudge factors in the correction of "$ab=ba$" to "$ab=ba[a,b]$." The bigger $G'$ is, the bigger the error level is. In the second case, while not every element commutes with every other element, there are elements that do - the central elements $a\in Z(G)$. One can form either the derived series or descending central series from commutators, and the ascending central series from the idea of collecting central elements.
(There are some sophisticated ideas behind commutators and centers being dual to each other - ideas like marginal and verbal subgroups, and varieties of groups. See Arturo's answer here.)
So $\gamma_1(G)=Z(G)$ represents the biggest "abelian" part $Z$ of $G$, where we change our idea of abelian from "all elements of $Z$ commute" to "all elements of $Z$ commute with all elements of $G$," so in reality we are measuring something tighter than abelianness (centrality), but it makes sense to do so because we may as well want this measure of abelianness to depend on how the subgroup interacts with the rest of the group $G$, not just with itself. If we change the focus on our camera lens, we may as well look at $G/Z(G)$: then the center here is $\gamma_2(G)/Z(G)$. In this way $\gamma_2(G)$ is a sort of "secondhand" center. We can continue on in this way to form the ascending series.
Observe $x\in\gamma_2(G)$ iff $[xZ(G),G/Z(G)]=1_{G/Z(G)}=Z(G)$ iff $[x,G]\subseteq Z(G)=\gamma_1(G)$. This relates centrality to commutators (since $x$ is central iff $[x,G]=1$). The idea behind any given central series $1=G_0\triangleleft G_1\triangleleft G_2\triangleleft\cdots\triangleleft G$ is that each $G_k$ contains the "error" from $G_{k+1}$ being central; this is encoded in the relation $[G_{k+1},G]\subseteq G_k$. This sort of telescopy encodes the original interpretation where we iteratively refocus our lens and zoom in on the central elements (or in reverse, refocus our lens according to the center, then zoom out for the next center).
