Homomorphism and Isomorphism

By homomorphism we mean a mapping from one algebraic system with a like algebraic system which preserves structures. Definition Let… Click here to read more

Definition If $$f$$ is a homomorphism of a group $$G$$ into a $$G’$$, then the set $$K$$ of all those… Click here to read more

Here’s some examples of the concept of group homomorphism. Example 1: Let $$G = \left\{ {1, – 1,i, – i}… Click here to read more

Definition Let $$G$$ and $$G’$$ be any two groups with binary operation $$ \circ $$ and $$ \circ ‘$$, respectively…. Click here to read more

Theorem 1: If isomorphism exists between two groups, then the identities correspond, i.e. if $$f:G \to G’$$ is an isomorphism… Click here to read more

Theorem 1: Cyclic groups of the same order are isomorphic. Proof: Let $$G$$ and$$\,G’$$ be two cyclic groups of order… Click here to read more

Cayley’s Theorem: Every group is isomorphic to a permutation group. Proof: Let $$G$$ be a finite group of order $$n$$…. Click here to read more

Example 1: Show that the multiplicative group $$G$$ consisting of three cube roots of unity $$1,\omega ,{\omega ^2}$$ is isomorphic… Click here to read more