Question 1 Question 2

a. By extension, for any field F, a n×n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n×n orthogonal matrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is
The characteristic polynomial of ρρ has degree 3 (since the space V has dimension 3). Now, every polynomial of degree 3 has a real root and it is by definition eigenvalue of ρρ. Further since you know that ρρ is rotation it does not change vector's length so absolute value of any eigenvalue has to be 1. So you have two cases - eigenvalue is 1 or -1. 
b. Let |G| ≥ 2 (possibly |G| = ∞). If G has no proper nontrivial subgroups, then G and hei are the only subgroups. Let a ∈ G be a nonidentity element. Then the subgroup generated by a cannot be hei, so hai = G, hence G is cyclic. If |G| = ∞, then G ∼= Z. But Z has nontrivial proper subgroups. Thus |G| < ∞. Suppose |G| = n. Since G is cyclic (that is a CRUCIAL point), for EVERY divisor d of n, ∃ a subgroup H of order d. If d 6= 1 or d 6= n, then H is a proper, nontrivial subgroup of G. Therefore, the only divisors of n are n and 1, hence n is prime.
c. We also define the restriction of this operator to the subspace Vd as ∆d : Vd → Vd−2 (for d ≥ 2). This restricted operator will be important, since we intend to study its kernel. A polynomial f : U → C, where U is some open subset of R 3 , that satisfies Laplaces equation ∆f = 0 (i.e. is in the kernel ker(∆)) is called...