1. There will be 6 problems: 2 easy, 2 intermediate, and 2 more challenging.
2. The topics will cover the material on group theory in the book discussed in the lecture/discussions, with focus on Section 4.4 and later sections.
3. There will be no question asking to reproduce a proof of a theorem from
the book, instead, the problems will ask to use the theory we went through
so far to
(a) solve some given problems about concrete groups (like D_2n, S_n, Z_n,
etc.)
(b) prove some simple general facts about groups.
The problems will be similar to homework problems and to recommended problems
suggested on the homework assignment website.
4. Here are some topics to focus on:
(a) Basic knowledge about concrete groups we went through:
D_2n, S_n, A_n, Z_n, V_4, Q_8
(b) Automorphism group Aut(G) of a group G, its basic properties, and connectins to group G
(c) Sylow theorem and its applications
(d) Direct products and recognizing direct products
(e) Applications of the fundamental theorem of finitely generated abelian groups; invariant factors, elementary divisors
(f) Semidirect products and their applications, with focus on constructing groups of a given order with certain specific properties, and classifying finite groups of small order
5. It will be important that you express yourself clearly and correctly. It
will be expected that you write your arguments rigorously, but include just the
right amount of information. Please take this into account when preparing for
the midterm.