Non-trivial nilpotent group has non-trivial center

Suppose $\;\gamma_n=1\;$ but $\;\gamma_{n-1}\neq 1\;$ (according to my definition, the correct one, and thus $\;G\;$ is of class $\;n\;$), then

$$\gamma_n:=[\gamma_{n-1},G]=1\iff \forall\,x\in\gamma_{n-1}\;\;and\;\;\forall\,g\in G\;,\;\;x^{-1}g^{-1}xg=1\iff xg=gx\implies$$

$$\implies \gamma_{n-1}\le Z(G)\implies Z(G)\neq 1$$

In case of finite groups: one can prove that finite nilpotent groups are precisely those groups that are a (internal) direct product of their Sylow subgroups: $G \cong P_1 \times \dots \times P_n$. Hence looking at the centers: $Z(G) \cong Z(P_1) \times \dots \times Z(P_n)$ and it is well-known that centers of $p$-groups are non-trivial.

Schwarz Lemma - like exercise

Evaluate $\int_0^1\frac{\ln x\ln(1-x)}{1+x^2}\ dx$

Improper integral of $\frac{x}{e^{x}+1}$

Any linear fractional transformation transforming the real axis to itself can be written in terms of reals?

Why does reflecting a point (x,y) about y=x result in point (y,x)?

Show $ f(x) = \sum_{n=1}^{\infty} \frac{nx}{n^3 + x^3}$ ,$\ g(x) = \sum_{n=1}^{\infty} \frac{x^4n}{(n^3 + x^3)^2}$ are bounded on $[0, \infty)$.

This algorithm will find a basis for the span of some vectors. How/why does it work?

Show that norm of matrix $A$ is given by the square root of the largest eigenvalue of $A^tA$

If $p\geq 5$ is a prime number, show that $p^2+2$ is composite.

Riemann integrable and continuous almost everywhere.

Intersection of hypercube and hyperplane - features of resulting polytope?

Equation with limit