{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Algebra Jan 2004

# Algebra Jan 2004 - Algebra Qualifying Examination January...

This preview shows pages 1–2. Sign up to view the full content.

Algebra Qualifying Examination January, 2004 Directions: 1. Answer all questions. (Total possible is 100 points.) 2. Start each question on a new sheet of paper. 3. Write only on one side of each sheet of paper. Policy on Misprints: The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem in such a way that it becomes trivial. Notes: 1. All rings are unitary. All modules are unitary. 2. is the rationals, the reals, the complexes, and the integers. Questions 1. (10 points) Determine all groups that have exactly 3 subgroups. 2. (15 points) Let Z ( G ) and Inn( G ) denote the center of the group G and the group of inner automorphisms of G respectively. (i) Prove that G/Z ( G ) is isomorphic to Inn( G ). (ii) Suppose that the group Aut( G ) of automorphisms of G is cyclic. Prove that G is abelian.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}