Algebra Qualifying Exam
March 23, 2013
Do all five problems.
1. Let G be a group and let Aut(G) denote the group of automorphisms of G.
Let
Inn(G) = {φ ∈ Aut(G) | ∃h ∈ G such that φ(g) = h−1 gh ∀g ∈ G}
be the subgroup of inner automorphisms. Show that Inn(G) / Aut(G).
2. Prove the Third Isomorphism Theorem: For any group G with normal
subgroups H / G and K / G, where H ⊆ K we have:
(a) K/H / G/H
(b) (G/H)/(K/H) ∼
= G/K
3. Let R, S denote commutative rings with unities. Let φ : R → S be a
homomorphism of rings. Prove that if J ⊂ S is a prime ideal then φ−1 (J)
is either equal to R or a prime ideal of R.
4. Let V be the vector space of upper triangular 2 × 2 matrices over R. Let
1 1
0 1
1 0
A=
, B=
, C=
0 0
0 1
0 1
(a) Show that (A, B, C) is a basis of V .
T
(b) Define an inner product
onV by hX, Y i = tr(XY ). Find the or1 2
thogonal projection of
onto the subspace spanned by A and B.
0 3
(Here tr(M ) denotes the trace of the matrix M .)
5. Let T : V → V be a linear map on an n-dimensional vector space V .
(a) Suppose v1 , v2 , . . . , vn are non-zero eigenvectors of T , associated with
distinct eigenvalues. Show that {v1 , v2 , . . . , vn } is linearly independent.
Conclude that this forms a basis.
(b) Suppose each vi is also an eigenvector of S : V → V (with not necessarily the same eigenvalues). Show that ST = T S.