The Subgroup Proposition

The evolution of Algebra, through its initial three stages of symbolisation, creation of procedures, and generalisation, led to its fourth stage of pure abstraction of number systems and their arithmetical operations in the form of Group Theory. The Subgroup Proposition, though very basic, has useful applications to cyclic groups, normal subgroups, quotient groups, homomorphisms, elementary linear groups, etc., and, thereby, offers new simple proofs of certain number-theoretic propositions. More significantly, it also captures the spirit and essence of Abstract Algebra in terms of axiomatic algebraic structures of rings, fields, etc. and concomitant mathematical approaches and techniques. These methods provide deeper understanding concerning roots of algebraic equations, besides showing the impossibility of the three classical construction problems of Greek Geometry. Furthermore, it throws light into cognition in Sciences and daily life through illustration of category- theoretic thinking of transforming problems in one mathematical domain into a more tractable problem in another domain, reprsenting the current fifth stage of Algebra.