M230 FALL 2018 MIDTERM INSTRUCTIONS
1. There will be 5 problems: 2 easy, 2 intermediate, and one more challenging.
2. The topics will cover the material in the book discussed in the
lecture/discussions all the way up to Section 4.4, including. Nothing from
Section 4.5 will be on the midterm.
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 group.
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) Homomorphisms and quotient constructions; isomorphism theorems
(c) Group actions and permutation representations
(d) Lagrange's theorem, counting with cosets, Cauchy's theorem, criteria
on normality of a subgrroup
(e) Group actions by left translation and conjugacy
(f) Class equation and its applications
(g) Calculation of normalizers, centralizers and centers.
(h) Conjugacy in S_n
(i) Automorphisms of a group: Connection between inner automorphisms
and centers.
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.
M230
Last Modified: October 31, 2018