View source for Algebraically closed group

{{MSC|20E|03C}}

''existentially closed group''

A group $G$ for which every finite system of equations soluble over $G$ is already soluble in $G$.  Every group can be embedded in in an algebraically closed group.  Such groups are simple and not finitely generated.  Every group with soluble word problem can be embedded in every algebraic group and conversely.  An algebraically closed group cannot have a recursive presentation.

Scott initially defined a group to be algebraically closed if it has the defining property for systems of equations and inequations and called a group "weakly algebraically closed" if this holds for systems of equations; it was proved by B.H. Neumann that the two properties are equivalent.  The term ''[[existentially closed]] group'' is also used.

====References====
* Scott, W.R. "Algebraically closed groups" ''Proc. Am. Math. Soc.'' '''2''' (1951) 118-121 {{DOI|10.1090/S0002-9939-1951-0040299-6 }} {{ZBL|0043.02302}}
* Neumann, B.H. ''A note on algebraically closed groups'' ''J. Lond. Math. Soc.'' '''27''' (1952) 247-249 {{DOI|10.1112/jlms/s1-27.2.247}} {{ZBL|0046.24802}}
* Higman, Graham; Scott, Elizabeth "Existentially closed groups" London Mathematical Society Monographs, New Series, '''3''' . Clarendon Press (1988) {{ZBL|0646.20001}}
* Ol’shanskij, A.Yu.; Shmel’kin, A.L. ''Infinite groups'' in "Algebra. IV: Infinite groups, linear groups", Kostrikin, A.I. (ed.); Shafarevich, I.R. (ed.); Gamkrelidze, R.V. (ed.) Encyclopaedia of Mathematical Sciences '''37''' Springer (1993) {{ZBL|0782.00033}} {{ZBL|0787.20001}}