Monthly Archives: December 2020

2 bucket case of Stirling’s numbers of 2nd kind

Posted on December 31, 2020 by Sumant Sumant

Stirling number of 2nd kind deals with the case of labelled balls and unlabeled buckets. The case of 2 buckets and n balls give a close formula

Posted in Combinatorics | Tagged | Leave a comment

Order of composition of permutations vs drawing of mappings.

Posted on December 31, 2020 by Sumant Sumant

One has to be careful how one composes function and how one uses shortcut to determine the composition using cycle notation.

Posted in Abstract Algebra | Tagged | Leave a comment

Connection between Fermat’s Little Theorem and Circular Permutations

Posted on December 31, 2020 by Sumant Sumant

There is a simple connection between Fermat’s Little Theorem and Circular Permutation. It can be summarized as number of ways to make an bead ring with colors. So that none of the ring has all the beads of the same … Continue reading →

Posted in Abstract Algebra, Number Theory, Proof, Proof by Construction | Tagged | Leave a comment

Alternating Group A3 is a Normal Subgroup

Posted on December 31, 2020 by Sumant Sumant

is a subgroup of in which all the cycles in the set are even permutations. Since every has terms and half of them are even and half of them are odd. In we have terms. Therefore in we have half … Continue reading →

Posted in Abstract Algebra | Tagged , | Leave a comment

Even and Odd Permutations

Posted on December 31, 2020 by Sumant Sumant

In group theory an even and odd permutation refers to the number of transpositions a cycle can be split into. If number of transpositions are even we call it an even permutations and if number of transpositions are odd we … Continue reading →

Posted in Abstract Algebra | Leave a comment

Symmetric Group S4

Posted on December 30, 2020 by Sumant Sumant
Posted in Abstract Algebra | Tagged | Leave a comment

On what set does the operation a*b=a+b+ab defines a group ?

Posted on December 29, 2020 by Sumant Sumant
Posted in Abstract Algebra, Proof | Tagged , | Leave a comment

Basic Cases of Continued Fractions

Posted on December 29, 2020 by Sumant Sumant
Posted in Number Theory | Tagged , | Leave a comment

Calculating (25/19) using equivalent Gauss’ lemma & Quadratic Reciprocity

Posted on December 29, 2020 by Sumant Sumant
Posted in Number Theory | Tagged | Leave a comment

The 3 Important Quadratic Residues

Posted on December 29, 2020 by Sumant Sumant
Posted in Abstract Algebra, Number Theory | Tagged , | Leave a comment