Algebraically closed group

2020 Mathematics Subject Classification: Primary: 20E Secondary: 03C [MSN][ZBL]

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