Category Archives: Abstract Algebra

Proof: Subgroup of a Cyclic Group is Cyclic

Posted on January 10, 2021 by Sumant Sumant

The main idea in this proof is to realize that each of the element of the subgroup is also generated by the generator of the group. Therefore we can find an element with the smallest positive exponent and then pick … Continue reading →

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Proof: Cyclic Groups Are Always Abelian

Posted on January 2, 2021 by Sumant Sumant
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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.

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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 →

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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 →

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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 →

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Symmetric Group S4

Posted on December 30, 2020 by Sumant Sumant
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On what set does the operation a*b=a+b+ab defines a group ?

Posted on December 29, 2020 by Sumant Sumant
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The 3 Important Quadratic Residues

Posted on December 29, 2020 by Sumant Sumant
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Proof: Quadratic Reciprocity Law

Posted on December 28, 2020 by Sumant Sumant
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