# Groups: Isomorphism

## What is a Group Isomorphism

• A group isomorphism is a bijective function `f` that preserves the group operation, meaning if `G` and `H` are groups and `f : G -> H` is a function, then:

1. For all `a, b` in `G`, `f(ab) = f(a)f(b)`.
2. `f` is a bijection (one-one and onto).
• If there exists an isomorphism between two groups, we say that the groups are isomorphic, denoted by `G` is isomorphic to `H` or `G ≈ H`.

• Isomorphic groups are structurally the same but their elements may be different. They have the same group properties, but not necessarily the same elements.

## Properties of Group Isomorphisms

• Being a bijection, isomorphisms preserve order elements and group order. If the order of an element `a` in `G` is `n`, the order of `f(a)` in `H` will also be `n`.

• Isomorphisms also preserve the identity element, i.e., if `e` is the identity in G then `f(e)` is the identity in `H`.

• Isomorphisms preserve inverses. If `a` is an element of `G` and `a'` is its inverse, then the inverse of `f(a)` in `H` is `f(a')`.

• The composition of two isomorphisms is also an isomorphism. If `f: G→H` and `g: H→K` are two isomorphisms, then their composition `g∘f: G→K` is also an isomorphism.

## Identifying Group Isomorphisms

• Finding an isomorphism involves identifying a function that maps each element of one group to an element in the other group while preserving the group operation.

• Tabular representation can be useful for identifying isomorphisms. The first step is to establish a one-to-one correspondence between the elements of the two groups. Then, construct tables for the groups and check if the structure is preserved.

• Make sure your function confirms to all properties of an isomorphism. If it fails to hold any one property, it is not an isomorphism.

## The Significance of Group Isomorphisms

• Group isomorphisms help us understand the structure of groups, which is a key concept in algebra. By identifying isomorphisms, we can explore properties that are preserved across different groups.

• They play an important role in abstract algebra and number theory by helping the simplification of complex structures, making them easier to study.

• Isomorphic groups appear in a wide variety of mathematical disciplines, including geometry, topology, and analysis.