# Definition A [[Types of Maps|bijective]] [[homomorphism]] is an isomorphism. The two isomorphic sets $X$ and $Y$ are written as $ X \cong Y $ [[Group Isomorphism|Group isomorphisms]] are special cases of isomorphisms that preserve the group structure. [[Lie Algebra Isomorphism|Lie algebra isomorphisms]] are a special case which preserve the Lie bracket. # Examples * All $n$-[[Dimension|dimensional]] [[Vector Space|vector spaces]] over a [[field]] $\mathbb{F}$ are isomorphic to $\mathbb{F}^n$. * All two-element [[group|groups]] are isomorphic to $\mathbb{Z}_2$ [[Cyclic Group of Order 2|(Z2)]].