D1 - group isomorphism

group isomorphism

let $G$ and $\bar{G}$ be two groups, so an isomorphism from $G$ to $\bar{G}$ is a one-to-one onto function $ϕ : G \to \bar{G}$ that preserves the group operation

an operation in $G$ translates to an operation in $\bar{G}$

ϕ (a b) = ϕ (a) ϕ (b) \forall a, b \in G

if there is an isomorphism from $G$ onto $\bar{G}$ , they are called isomorphic:

G ≅ \bar{G}

identity

let $e$ and $\bar{e}$ be identities of $G$ and $\bar{G}$ , then:

ϕ (e) = \bar{e}

subgroup

if $K \leq G$ then $ϕ (K) \leq \bar{G}$