Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.
== Proof ==
This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. Specifically, x and y in G are related if and only if there exists h in H such that x = yh. If we can show that all cosets of H have the same number of elements, then each coset of H has...
In mathematics, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a ring.
Maxim Kontsevich and Don Zagier (2001) gave a survey of periods and introduced some conjectures about them.
== Definition ==
A real number is called a period if it is the difference of volumes of regions of Euclidean space given by polynomial inequalities with rational coefficients. More generally a complex number is called a period if its real and imaginary parts are periods.
Periods are numbers that...
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Q: Can a nowhere differentiable Jordan curve pass only through points with both irrational coordinates?
Suppose that $J$ is Jordan curve with the following properties:
1) $J$ is nowhere differentiable, that is, the injective continuous map $\phi$ mapping $[0,1]$ into $\mathbb R^2$ such that $\phi(0)=\phi(1)$ and the restriction of $\phi$ to $[0,1)$ is injective is nowhere differentiable.
2) $J$ p...
