Abstract
This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.
 
Highlights
One of the two greatest achievement of group theory inventing is solving equations
Euler Formula is given by einθ = cosnθ + i sin nθ and De Moivre Theorem states that zm = rmeimθ =
The definition of group was followed by Cayley Theorem due to Cayley (1821-1895)
Summary
One of the two greatest achievement of group theory inventing is solving equations. The other one is ‘counting’. Since they are P-Factor Groups, they have normal P-Sylow Subgroups. Because they all have index 2 in their PGroups, they are the maximal proper normal P-Sylow Subgroups and their factor groups are abelian accounting to the solubility of xnm ± 1 by composition series. Cauchy stated and proved that the group needs not to be abelian so far it is finite of prime order.
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