Residually Finite Groups

Definitions and mathematical statement

A group $G$, with identity element $1_G$, is residually finite if \[ \forall g\in G\setminus\{1_G\}\ \exists\text{ a finite group }F\ \exists\text{ a group homomorphism }\varphi:G\to F \quad \varphi(g)\ne1_F. \] Equivalently, each nonidentity element survives in some finite quotient. Under the residual-finiteness interpretation supplied by the title and background, the question "Is every group well-approximated by finite groups?" is the universal assertion \[ \forall\text{ groups }G\quad G\text{ is residually finite}. \] No finite-generation or finite-presentation hypothesis occurs in this assertion.

The supplied literature triage dated 2026-08-17 classifies this universal assertion as false under that interpretation, identifying nontrivial infinite simple groups as a counterexample class and citing Higman (1951). The registry's separate status field is open; this is retained as source metadata rather than a mathematical conclusion. The background speculates about finitely generated or finitely presented variants, but does not actually formulate one; in particular it cannot silently replace the universal domain above. The supplied Higman reference is described by the triage as providing a finitely generated infinite simple group, so finite generation by itself is not an asserted escape from that counterexample class.

If "well-approximated" means an approximation notion other than residual finiteness, the source does not define it. Such a reading would be a different mathematical question and is not resolved by the conditional formulation given here.

Short English statement

Under the source's residual-finiteness interpretation, every nonidentity element would have to remain nonidentity in some finite image, for every group. The supplied 2026-08-17 triage reports this universal assertion false using infinite simple groups. An alternative meaning of finite approximation is not specified.

Sources

• Local UnsolvedMath registry, permanent ID 152, source code catalogue entry, "Residually Finite Groups," including the supplied literature triage dated 2026-08-17. Membership in Ben Green's problem set is registry metadata; no direct Green problem reference is supplied for this wording.
• G. Higman, "A Finitely Generated Infinite Simple Group," Journal of the London Mathematical Society 26 (1951), 61--64, supplied primary reference: https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/jlms/s1-26.1.61. The supplied triage cites this for the counterexample class under the residual-finiteness reading.