  We study the stellar and gas kinematics of the brightest group galaxies
(BGGs) in dynamically relaxed and unrelaxed galaxy groups for a sample of 154
galaxies in the SAMI galaxy survey. We characterize the dynamical state of the
groups using the luminosity gap between the two most luminous galaxies and the
BGG offset from the luminosity centroid of the group. We find that the
misalignment between the rotation axis of gas and stellar components is more
frequent in the BGGs in unrelaxed groups, although with quite low statistical
significance. Meanwhile galaxies whose stellar dynamics would be classified as
`regular rotators' based on their kinemetry are more common in relaxed groups.
We confirm that this dependency on group dynamical state remains valid at fixed
stellar mass and Sersic index. The observed trend could potentially originate
from a differing BGG accretion history in virialised and evolving groups.
Amongst the halo relaxation probes, the group BGG offset appears to play a
stronger role than the luminosity gap on the stellar kinematic differences of
the BGGs. However, both the group BGG offset and luminosity gap appear to
roughly equally drive the misalignment between the gas and stellar component of
the BGGs in one direction. This study offers the first evidence that the
dynamical state of galaxy groups may influence the BGG's stellar and gas
kinematics and calls for further studies using a larger sample with higher
signal-to-noise.
