# Tag Archives: Algebra

## Schur’s Lemma

Schur’s Lemma is a useful theorem in algebra that is surprisingly easy to prove. Schur’s Lemma: Let and be right -modules and let be a nonzero -module homomorphism. (i) If is simple, then is injective. (ii) If is simple, then is … Continue reading

## N is a simple nonzero right A-module (Equivalent Conditions)

Let be a nonzero right -module. Then the following are equivalent: (i) is simple. (ii) for all (iii) for some maximal right ideal of . Proof: (i)=>(ii) Let . is a submodule of . (Let and , then , ). … Continue reading

## Opposite Algebra

Let be an -algebra. Define an operation by . This algebra is called the opposite algebra to . We verify that it forms an -algebra. Bilinearity comes from the following computations: Associativity is true from The unity element is the … Continue reading

## Center of Matrix Algebra / Matrix Ring is Scalar Matrices

We will prove that the center , where is an -algebra (or ring with unity). One direction is pretty clear. Let . Then for some . , , so is in the center . The other direction will require the … Continue reading

## Conjugacy Classes of non-abelian group of order p^3

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## S_n has Trivial Center for n greater than 3

We will prove that for , . Proof: Clearly, . Let . There exists distinct elements of such that . Consider the transposition , where is distinct from . (Since , such a exists.) Then, . Therefore, Do check out … Continue reading

## Singapore Haze & Subgroup of Smallest Prime Index

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## Proof of Wilson’s Theorem using Sylow’s Theorem

Wilson’s theorem is a useful theorem in Number Theory, and may be proved in several different ways. One of the interesting proofs is to prove it using Sylow’s Third Theorem. Let , the symmetric group on p elements, where p … Continue reading

Quadratic Equations Questions and Solutions Today we will discuss about the quadratic equation . The way to derive the Quadratic Formula is to use completing the square. First, we divide throughout by a, where a is assumed to be non-zero … Continue reading

## Difference of Cubes Video

A geometric interpretation of the Difference of Two Cubes formula.

## Make (y) the subject of the formulae (O Level Maths Tuition)

Question: Make (y) the subject of the formulae (1) Solution: Cross multiply,