Φ(G) and Φ'(G) Coincidence

Definitions and mathematical statement

The literal imported question is \[\Phi(G)\stackrel{?}{=}\Phi'(G).\] The supplied statement does not define either side, and therefore this expression alone is not a well-posed mathematical assertion. The supplied literature triage identifies the intended source as Problem 57 of B. Green's 100 Open Problems: there $G$ is an abelian group, $\Phi(G)$ is a convex hull of generalized convolutions on $G$, and $\Phi'(G)$ imposes the condition that one factor depend only on $x_1+x_2$. The exact generalized-convolution formula, admissible factors, underlying function classes, averaging or measure conventions, and convex-hull conventions are not supplied here. Consequently the question can be retained only as the source-defined function-space equality, pending those definitions; no guessed definition of either space is introduced.

The original background's interpretation as equality of Frattini-like subgroups is explicitly identified as erroneous by the supplied review, which also calls the imported algebra category substantively wrong. It is not used as a definition. The review, checked 2026-08-17, links the intended problem to generalized convolution algebras and $\mathrm{UAP}^{2}$ functions, reports that Green conjecturally expects the spaces not to coincide, and reports no resolution. It says the intended source problem remains open, while no status can be assigned to the literal undefined statement. Its PARTIAL-PROGRESS label concerns this source diagnosis, not a theorem about Frattini subgroups.

Short English statement

Ask whether Green's two source-defined spaces of functions on an abelian group are equal. The local record omits their exact definitions and contains an erroneous Frattini-subgroup explanation, so the intended function-space question needs source definitions before a precise formulation is possible.

Sources

• Local UnsolvedMath registry, permanent ID 118, source code catalogue entry, "$\Phi(G)$ and $\Phi'(G)$ Coincidence"; supplied statement, background, and corrective literature triage.
• B. Green, 100 Open Problems, Problem 57, version retrieved 2026-08-17 according to the supplied maintained-tracker reference: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The review identifies this problem as the intended source and describes its generalized-convolution and $\mathrm{UAP}^{2}$ context.
• Supplied literature triage, checked 2026-08-17, partially solved / PARTIAL-PROGRESS. It records that the local code corresponds to source Problem 57 only after source consultation, and that no silent repair was made; dated triage does not exclude later or unindexed results.