Besides studying groups by decomposing them into substructures, it's also possible to study group by building a special map between two groups that preserves some group structure.Consider the homogeneous matrix group under multiplication and the non-zero real number under multiplication . It's possible to draw a map defined by , and there's when , which means...(read more)

Group theory, which studies a type of algebraic structures called group, is the very central and fundamental concept of algebra. By studying group theory, we build up some concepts and proofs with which we could use to drill deeper into the mathematical world.A group is a set equipped with a binary operation ¹ such that:For convenience, we usually omit the binary operation...(read more)

For two integers and , their greatest common divisor is the greatest number such that . Otherwise we could take such that is the common divisor of but is greater than , which is a contradiction to our previous definition.When evaluating greatest common divisor between and...(read more)

Prime numbers are the fundamental of number theory, stated by the fundamental theorem of arithmetic, that each positive integer except can be decomposed into the product of one or more primes.The proof of the theorem is intuitive and easy to prove, given a number , you first iterate through the primes less than or equal to , and divide by each primes until they are coprime, repeat the process until is...(read more)