isomorph

Isomorphism

Let $(G,\circ ),(H,\star )$ be groups. A function $\varphi :G\to H$ is an isomorphism if:

1. $\varphi$ is a homomorphism
2. $\varphi$ is bijective

If any such an isomorphism exists, we say $G$ and $H$ are isomorphic, written $G\cong H$.

Difference to Homomorphisms

A homomorphism only requires structure preservation ($\varphi (x\circ y)=\varphi (x)\star \varphi (y)$).
An isomorphism additionally requires that $\varphi$ is bijective, meaning:

• Every element in $H$ is reached (surjectivity)
• Different elements in $G$ map to different elements in $H$ (injectivity)

An isomorphism is a homomorphism that has an inverse homomorphism. Thus, isomorphic groups are structurally indistinguishable – they are essentially "the same group" written in different notation.

Properties of Isomorphic Groups

If $G\cong H$, then:
$G$ and $H$ have the same order (cardinality).
Every structural property is shared: commutativity, cyclic nature, order of elements, …
The groups are identical up to renaming of elements.

$({\mathbb{Z}}_4,+,\bar{1})\cong (\{1,i,-1,-i\},\cdot ,i)$ as both are cyclic groups of order 4, although their elements and operations appear different.

For an isomorphism $\varphi :G\to H$:

$\ker (\varphi )=\{e_G\}$ (equivalent to injectivity)
$\text{im}(\varphi )=H$ (equivalent to surjectivity)
${\varphi }^{-1}:H\to G$ exists and is also an isomorphism