Intereting Posts

Find all Integral solutions to $x+y+z=3$, $x^3+y^3+z^3=3$.
What operations can I do to simplify calculations of determinant?
generalized inverse of a matrix and convergence for singular matrix
Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$.
Another counting problem on the number of ways to place $l$ balls in $m$ boxes.
Isomorphism between quotient rings of $\mathbb{Z}$
How to integrate $ \int_0^\infty \sin x \cdot x ^{-1/3} dx$ (using Gamma function)
Solving Induction $\prod\limits_{i=1}^{n-1}\left(1+\frac{1}{i}\right)^{i} = \frac{n^{n}}{n!}$
Prove $\sum_{k=0}^{58}\binom{2017+k}{58-k}\binom{2075-k}{k}=\sum_{k=0}^{29}\binom{4091-2k}{58-2k}$
Mathematical structures
The convergence of a sequence with infinite products
Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$
Simplifying $\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}$
How to solve $4\sin \theta +3\cos \theta = 5$
Coupon Collector Problem – expected number of draws for some coupon to be drawn twice

Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group *without* using Lagrange’s theorem? (Or, equivalently, that the order of the group is an exponent for every element in the group?)

The simplest proof I can think of uses the coset proof of Lagrange’s theorem in disguise and goes like this: take $a \in G$ and consider the map $f\colon G \to G$ given by $f(x)=ax$. Consider now the *orbits* of $f$, that is, the sets $\mathcal{O}(x)=\{ x, f(x), f(f(x)), \dots \}$. Now all orbits have the same number of elements and $|\mathcal{O}(e)| = o(a)$. Hence $o(a)$ divides $|G|$.

This proof has perhaps some pedagogical value in introductory courses because it can be generalized in a natural way to non-cyclic subgroups by introducing cosets, leading to the canonical proof of Lagrange’s theorem.

- Show that order of $a^k$
- Semisimple ring problem
- Any quotient group of $(\mathbb Q,+)$ is torsion
- Field of sets versus a field as an algebraic structure
- Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$
- How to prove that $z\gcd(a,b)=\gcd(za,zb)$

Has anyone seen a different approach to this result that avoids using Lagrange’s theorem? Or is Lagrange’s theorem really the most basic result in finite group theory?

- Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$
- Localization of a valuation ring at a prime is abstractly isomorphic to the original ring
- Existence of normal subgroups for a group of order $36$
- Checking the maximality of a principal ideal in $R$
- Question about the Euclidean ring definition
- Binary operation (english) terminology
- Kummer extensions
- Minimal polynomial of $\sqrt{2} + \sqrt{3}$
- Find all ring homomorphisms from $\Bbb Q$ to $\Bbb R$
- What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?

Consider the representation of $\langle a \rangle$ on the free vector space on $G$ induced by left multiplication. Its character is $|G|$ at the identity and $0$ everywhere else. Thus it contains $|G|/|\langle a \rangle|$ copies of the trivial representation. Since this must be an integer, $|\langle a \rangle|$ divides $|G|$. Developing character theory without using Lagrange’s theorem is left as an exercise to the reader.

- Matrices that Differ only in Diagonal of Decomposition
- Tricky proof that the weighted average is a better estimate than the un-weighted average:
- How can a social welfare function be a linear combination of von Neumann-Morgenstern utility functions?
- No function that is continuous at all rational points and discontinuous at irrational points.
- Reflection of a curve around a slant line
- Why are properties lost in the Cayley–Dickson construction?
- Natural logarithms base $e$
- Prove that Gauss map on M is surjective
- Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
- Do eigenvalues of a linear transformation over an infinite dimensional vector space appear in conjugate pairs?
- Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$
- Expectation of the maximum of i.i.d. geometric random variables
- Half order derivative of $ {1 \over 1-x }$
- What axioms need to be added to second-order ZFC before it has a unique model (up to isomorphism)?
- Proof of Cohn's Irreducibility Criterion