NORMAL SUBGROUPS AND QUOTIENT STRUCTURES OF ORTHOGONAL RHOTRIX GROUP

This paper studies the structure of the non-commutative orthogonal rhotrix
group ORn(F) through its normal subgroups and quotient groups. The triv
ial subgroup and the special orthogonal rhotrix group SORn(F), arising as
the kernel of the determinant homomorphism, are shown to be normal, yield
ing baseline and orientation-based quotients. The centre CORn(F) is estab
lished as a normal subgroup of ORn(F), and its interaction with SORn(F)
is analyzed in the non-commutative setting. While CORn(F) need not be
contained in SORn(F), the intersection CORn(F)∩SORn(F) is always nor
mal in SORn(F), leading to well-defined quotient that captures the intrinsic
non-abelian structure of the group. Under a sufficient determinant condition
on central elements, the stronger quotient SORn(F) CORn(F) is obtained.
These results provide a canonical quotient-based decomposition of ORn(F).