MASTER’S EXAM, ALGEBRA, AUGUST 2011
1. A matrix Ais said to be idempotent if A2=A. Show that every finite, real,
symmetric idempotent matrix represents a linear transformation which is an
orthogonal projection and identify the subspace onto which it is projecting.
2. Prove that any set of 3 vectors in R2is linearly dependent.
3. State and prove the Rank Plus Nullity theorem for Rn.
4. Let Gbe a group. Show that if each element of Ghas order 2, then Gis abelian.
5. If Gis a group, show that if His a normal subgroup of G, then the multiplication
of right cosets Hg1H g2=Hg1g2is well defined.
Find an example of a group Gwith a subgroup Hfor which this multiplication
would not be well defined.
6. Let Gbe a group of order 245. Show that Ghas a normal subgroup of order
49.
7. Find an ideal in Z×Zthat is prime but not maximal.
8. Let Rbe a commutative ring with 1 6= 0. Suppose Iand Jare ideals of Rsuch
that I+J=R. Prove that
R/(I∩J)∼
=R/I ⊕R/J.
9. Prove that the Frobenius map on a field of finite characteristic is always injective
and give an example of a field of finite characteristic for which it is not surjective.
10. Prove that the order of a finite field is a power of a prime.