Preprint Classifies Rota–Baxter Operators on Two Finite Groups

A preprint dated 28 August 2026 reports a classification of Rota–Baxter operators on two finite groups. On E3, every operator is reduced to one of four named families: B1, B2x, B3x or B4l,y. On H3, every non-trivial operator is reduced to one of the listed representatives. Both reductions use the manuscript's stated equivalences, including automorphism conjugation and an additional action on operators. In this setting, an orbit is the collection of operators connected by those allowed transformations.

This is not a participant study. The unit of analysis is an algebraic group G together with a map $B:G\to G$, rather than participant observations.

The rule behind the list

At the heart of the work is the identity that defines a group Rota–Baxter operator. The map must satisfy $B(g)B(h)=B(gB(g)hB(g)^{-1})$ for all group elements g and h. In plain language, the product of the two outputs on the left must equal the operator applied to the conjugated argument on the right. The inverse notation denotes a group inverse.

The main setting is the finite Heisenberg group H3 (Zp) and the extraspecial group E3 of order p3. Here p is the group parameter, restricted to at least 3.

A center that stays put

One theorem covers every Rota–Baxter operator on an extraspecial group. It says that the operator maps the group's center into itself, written $B(Z(G))\subseteq Z(G)$. Applying B to elements of the center therefore produces elements that remain in the center.

The paper gives explicit coordinate formulas for operators on H3 and E3. The first two coordinate maps are linear in t and s. The central-coordinate map combines linear terms, a mixed quadratic term in t and s, separate quadratic terms in t and s, and a term linear in r. The permitted parameters come with group-specific restrictions.

Turning formulas into families

For E3, the classification has four named families. Every operator is reduced to B1, B2x, B3x or B4l,y under automorphism conjugation and the paper's additional action, and the listed families occupy different orbits.

For H3, the theorem covers every non-trivial operator and reduces it to one of the listed representatives under automorphism conjugation and the paper's additional action.

When bijectivity is possible

A separate result identifies when an H3 operator and its associated transformed map are both bijective, meaning each maps the group one-to-one and onto itself. Up to the stated equivalences, this happens exactly in the B1, B3 and O1 families.

A further corollary asserts that, for an extraspecial group of exponent p, there exist Rota–Baxter operators for which both the operator and its associated transformed map are bijective.

The paper also gives an if-and-only-if construction criterion for groups of nilpotency class 2. It considers maps of the form $B(g)=\theta(g)\phi(g)$, where g is a group element, theta takes values in the center and phi is an automorphism of the group. The condition on theta is necessary and sufficient for the constructed map to be a Rota–Baxter operator. The class-2 assumption is expressed as $[[G,G],G]=\{e\}$, meaning that the third-level commutator subgroup is the identity subgroup; G is the group, [ , ] is the commutator operation and e is the group identity.

Another class, another list

The paper also classifies non-trivial splitting operators, identifying specific representatives on H3 and E3 up to automorphism conjugation and the additional action used in the classification.

What the classification does not cover

The scope is specific. The orbit classifications concern E3 and the non-trivial operators on H3, while the broader extraspecial-group results concern center preservation and selected constructions and existence statements. In E3, different choices of the listed parameters can define the same Rota–Baxter operator, so the parameterization is not unique.

The manuscript is labeled arXiv:2608.28291v1 and dated 28 Aug 2026. Its analysis concerns algebraic groups and maps rather than participant observations.

Paper data and sources

Original title: Rota---Baxter operators on extraspecial groups
Authors: Andrey Savelyev
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text