Brightest group galaxies in COSMOS-Web: Evolution of the size–mass relation since z = 3.7

We present the first comprehensive study of the structural evolution of brightest group galaxies (BGGs) from redshift (z ≃ 0. 08) to (z = 3. 7) using the James Webb Space Telescope 's 255-hour COSMOS-Web program. This survey provides deep NIRCam imaging in four filters (F115W, F150W, F277W, and F444W) in (∼ 0. 54 deg ²), allowing robust size and morphological measurements for (∼ 1700) BGGs spanning (∼ 12) Gyr of cosmic history. High-resolution imaging enables consistent measurement of galaxy sizes in the rest-frame optical (red to near-infrared; ∼6000–8000, Å) across cosmic time through redshift-dependent filter selection. We classified BGGs as star-forming and quiescent using both rest-frame NUV-- (r) -- (J) colors and redshift-dependent specific star formation rate (sSFR) thresholds. Our structural analysis reveals that quiescent BGGs are systematically more compact than their star-forming counterparts across all redshifts, exhibiting steeper size--mass slopes ( (α_ QG ∼ 0. 6) --1. 2 vs. (α_ SF ∼ 0. 0) --0. 3). The effective radius evolves as (Rₑ ∝ (1+z) ^ -α), with (α = 0. 96 ± 0. 07) for star-forming BGGs and (α = 1. 24 ± 0. 09) for quiescent BGGs, indicating stronger size growth in quenched systems. The corresponding growth factor at fixed stellar mass ( (łog M_* = 10. 7) ) from (z = 3. 7) to (z = 0. 08) is (∼ 4. 4) for star-forming and (∼ 6. 6) for quiescent BGGs. The intrinsic scatter in the size--mass relation increases toward higher redshift for both populations, reaching (∼ 0. 3) --0. 4 dex at (z > 2), reflecting greater structural diversity in the early universe. Compared to field galaxies, BGGs show systematically smaller sizes at fixed stellar mass, particularly among quiescent systems, highlighting environmental effects on galaxy structure. We further compare the evolution of the quiescent fraction, the Sérsic index, and ellipticity with those of field galaxies, finding consistent trends that reinforce our main conclusions. These results establish the foundation for understanding how group-scale environments shape the structural evolution of central galaxies and provide crucial constraints for models of galaxy formation in intermediate-mass dark matter halos.