The lower central series of a group $G$ is the sequence $G = \gamma_1(G) \supseteq \gamma_2(G) \supseteq \cdots$ with $\gamma_{i+1}(G) = [\gamma_i(G), G]$. The group is nilpotent if and only if this series reaches the trivial subgroup.

Two vectors of an inner product space are orthogonal when their inner product is zero.

Build a square on each side of a right triangle. The two small squares together have the same area as the big square on the longest side. For the sides $3$, $4$ and $5$: $9 + 16 = 25$.

The operator norm of a continuous linear map measures the largest factor by which the map can stretch a vector. It makes the space of continuous linear maps into a normed space.

The theorem lists all the finite simple groups. Most of them come in a few infinite families, and only $26$ are special cases. Since each finite group is built from simple groups, this list is a map of all the building blocks.

Suppose that $f$ does not change in any direction at any point, and that the domain is in one piece. Then $f$ cannot change at all. On a domain in two pieces, $f$ can take a different constant value on each piece.

In an infinite set you can always pick one more element. Picking forever gives a sequence of different elements, which is a copy of $\mathbb{N}$ inside the set.

The Taylor polynomial of degree $k$ of a function at a point has the same value and the same first $k$ derivatives there. It approximates the function near the point. The node Taylor polynomial gives the precise definition.

A power series can be differentiated like a polynomial, one term at a time. The new series converges on the same open interval.

An arithmetic function is a function on the positive integers with complex values, often one that expresses an arithmetic property of $n$. The definition of an arithmetic function gives the precise form.

A multilinear map is a map of several vector variables that is linear in each variable when the other variables are fixed.

For integers, the theorem says that a number modulo $15$ is the same information as the number modulo $3$ and modulo $5$. For each pair of remainders, exactly one class modulo $15$ has them. The ring version says the same for any family of pairwise coprime ideals.

To compute a line integral, write the curve with a parameter $t$. Then $ds$ becomes $\|\gamma'(t)\| \, dt$ and $d\mathbf{r}$ becomes $\gamma'(t) \, dt$, and the line integral is an ordinary integral in $t$.

A free abelian group is an abelian group with a basis. Each element is an integer combination of finitely many basis elements in exactly one way. The free abelian group of finite rank $n$ is isomorphic to $\mathbb{Z}^n$.

If the bottom part and the top part of a group can both be built from abelian layers, then so can the whole group. You stack the layers of the top part on the layers of the bottom part. For example, $S_4$ is built from $V$, $A_4/V$ and $S_4/A_4$.

A finite extension is a field extension of finite degree. The definition of a finite extension states it.

Along an eigenvector, the matrix $A$ acts as multiplication by a number. So on this line the system is the simple equation $y' = \lambda y$, with solution $e^{\lambda t}$. Write the start vector in a basis of eigenvectors, and let each part grow or decay at its own rate.

A subset of a metric space is totally bounded when, for each size, it can be covered by finitely many pieces of that size. Compact sets are exactly the totally bounded sets that are complete.

In $n$ trials with success probability $p$, you expect $np$ successes. For $100$ tosses of a fair coin, the expected number of heads is $50$.

A walk in a graph is a sequence of vertices in which consecutive vertices are adjacent. Walks are the basic notion of movement in a graph, and trails, paths, circuits and cycles are special walks. The node walk gives the precise definition.