Final answer:
Oₙ, the orthogonal group, is a normal subgroup of Iso(Rⁿ), the group of isometries in Rⁿ, because it is closed under the group operations, and it is invariant under conjugation by elements of Iso(Rⁿ).
Step-by-step explanation:
To prove that On (the orthogonal group) is a normal subgroup of Iso(Rn) (the group of isometries in Rn), we must show two things: that On is a subgroup of Iso(Rn), and that it is invariant under conjugation by elements of Iso(Rn).
First, the orthogonal group On is indeed a subgroup of Iso(Rn) since it consists of all linear isometries that preserve distances and angles in Rn, and these are clearly isometries. The identity element is the identity matrix, which is an orthogonal matrix, and taking the inverse of an orthogonal matrix yields another orthogonal matrix, so it is closed under inverses. The composition of two orthogonal matrices is also an orthogonal matrix, thus it's closed under composition.
To prove normality, we take any isometry f in Iso(Rn) and an orthogonal transformation O in On and consider the conjugation fOf-1. Since f and f-1 are isometries, and O is an orthogonal transformation, the conjugate will also be an orthogonal transformation because it will also preserve distances. Therefore, fOf-1 will be in On, showing that On is a normal subgroup of Iso(Rn)