All we've studied so far is treating a group as a complete black box, discussing theories applicable to all kinds of groups, and might be really dull.In this text, we discuss a class of group called cyclic group, based on which we introduce an effective method called cyclic subgroups for ripping down a group and inspecting what kind of structure can a group possesses.We will apply the method of cyclic subgroups to...(read more)

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)