English
Let f : G →* H × I with surjective projections. If the image satisfies a line-condition, then the image is the graph of a monoid isomorphism.
Русский
Пусть f : G →* H × I с сюръективными проекциями. Если образ удовлетворяет условию по линиям, то образ есть граф изоморфизма.
LaTeX
$$$\\exists e : H \\simeq^* I,\\; \\operatorname{range} f = e.toMonoidHom.mgraph$$$
Lean4
/-- The graph of a group homomorphism as a subgroup.
See also `MonoidHom.mgraph` for the graph as a submonoid. -/
@[to_additive /-- The graph of a group homomorphism as a subgroup.
See also `AddMonoidHom.mgraph` for the graph as a submonoid. -/
]
def graph (f : G →* H) : Subgroup (G × H) where
toSubmonoid := f.mgraph
inv_mem' {x} := by simp +contextual
