Algebra Qualifying Exam
September 13, 2012
Do all five problems.
1. Let G be a group. A subgroup H is said to be normal if gHg −1 = H for
every g ∈ G. Show that it is enough to show that gHg −1 ⊆ H for every
g ∈ G.
2. If H and K are normal subgroups of a group G with HK = G, prove that
G/(H ∩ K) ∼
= (G/H) × (G/K).
Note that you may apply the First Isomorphism Theorem but not the
Second or Third.
3. Let R be an integral domain (with multiplicative identity 1 6= 0) and let Q
be its field of fractions. [Recall that Q is defined to be set of equivalence
classes ab where a ∈ R, b ∈ R \ {0}; here the equivalence relation is defined
so that ab = dc if and only if ad = bc and addition and multiplication are
defined as usual for fractions.]
Prove: if σ : R → R is an automorphism then there is a unique automorphism σ : Q → Q such that
σ
r
1
=
σ(r)
for all r ∈ R.
1
4. Let V be a finite-dimensional vector space and let T : V → V be linear.
Prove that if rank(T ) = rank(T 2 ), then V = range (T ) ⊕ null (T ).
5. Let V denote the vector space (over R) of polynomials of degree ≤ 2 with
real coefficients. Define T : V → V by
T (p)(x) = p(x) + 3xp0 (x) + 4p00 (x).
Then
(a) Find [T ]β , the matrix representation with respect to the basis β =
(1, x, x2 ).
(b) Is the transformation T diagonalizable? If so express [T ]β as a product QDQ−1 where D is a diagonal matrix, otherwise explain why T
is not diagonalizable.