This post collects the top 5 most popular posts according to Google search console.

Read moreThis post collects the top 5 most popular posts according to Google search console.

Read moreThe Structure of SL_2(F_3) as a Semidirect Product

In this post we determine $SL_2(\mathbb{F}_3)$ using Sylow theory and linear algebra.

Read moreA Separable Extension Is Solvable by Radicals Iff It Is Solvable

We show that a separable extension is solvable by radical iff it is solvable, i.e. it has a Galois closure with solvable Galois group. The proof is done in a general setting.

Read morePicard's Little Theorem and Twice-Punctured Plane

We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.

Read moreSL(2,R) As a Topological Space and Topological Group

In this post we show that $SL(2,\mathbb{R})$ can be identified as the inside of a solid torus and see what we can learn from it.

Read moreArtin's Theorem of Induced Characters

We give a relatively more detailed proof of Artin's theorem in representation theory of finite groups as well as an example of dihedral group.

Read moreChinese Remainder Theorem in Several Scenarios of Ring Theory

We study the Chinese remainder theorem in various contexts and abstract levels.

Read moreProjective Representations of SO(3)

In this post we study projective representations of $SO(3)$, although we will make more use of $SU(2)$. At the end of this post we reach the conclusion that one will think about polynomials with odd or even terms. Projective representations have its own significance in physics although the room of this post is too small to contain it. Nevertheless, the reader is invited to use linear algebra much more extensively with a taste of modern physics in this post.

Read moreIn this post we deliver the basic computation of the quadratic reciprocity law and see its importance in algebraic number theory.

Read moreVague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions

We give an introduction to vague convergence and see several equivalent conditions of it.

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