# Algebra Qualifying Exam September 15, 2011 Do all five problems.

```Algebra Qualifying Exam
September 15, 2011
Do all five problems.
1. Let G be any group.
(a) Let φ : G → G be defined by φ(g) = g 2 for all g ∈ G. Prove that φ
is a homomorphism if and only if G is abelian.
(b) If G is abelian and finite, show that φ is an automorphism if and
only if G has odd order.
2. Let G be a group and let S = {xyx−1 y −1 | x, y ∈ G}. Prove: If H is a
subgroup of G and S ⊆ H, then H is a normal subgroup of G.
3. Prove that in an integral domain D every prime element is an irreducible.
4. Find necessary and sufficient conditions on α, β, γ ∈ R such that the
matrix


1 α β
 0 0 γ 
0 0 1
is diagonalizable over R.
5. Let Mn (R) be the vector space of n &times; n matrices with entries in R and
let S and Z denote the set of real n &times; n symmetric and skew-symmetric
matrices, respectively (recall that an n &times; n matrix A is skew-symmetric if
AT = −A).
(a) Show that dim(S) =
n(n+1)
.
2
A brief justification is sufficient.
(b) Let T : Mn (R) → Mn (R) be the linear transformation defined by
T (A) =
A + AT
2
for all A ∈ Mn (R). Prove that ker(T ) = Z and im(T ) = S.
(c) Compute dim(Z).
```