This course is an exercise session for "Introduction to Algebra II'' (ZUA.A203). The materials for exercise are chosen from that course.
To become familiar with important notions such as the axiom of groups, subgroups, residue classes, order, cyclic groups, symmetric groups, homomorphisms of groups, normal subgroups, the fundamental theorem on group homomorphisms, conjugacy classes, class equation, and actions of groups.
To become able to prove by him/herself basic properties of these objects.
group, subgroup, residue class, order, cyclic group, symmetric group, homomorphism of groups, normal subgroup, the fundamental theorem on group homomorphisms, conjugacy class, class equation, action of a group
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Students are given exercise problems related to what is taught in the course "Introduction to Algebra II'''.
Course schedule | Required learning | |
---|---|---|
Class 1 | Discussion session on the axiom of groups, typical examples of groups, first properties of groups | Details will be announced during each lecture. |
Class 2 | Discussion session on basic properties of the operation in a group and of the identity and inverse elements | |
Class 3 | Discussion session on the definition of a subgroup, criterion for subgroups, and examples of subgroups | |
Class 4 | Discussion session on right- and left-cosets by a subgroup | |
Class 5 | Discussion session on the order of a group and Lagrange's theorem | |
Class 6 | Discussion session on the order of an element of a group and on cyclic groups | |
Class 7 | Discussion session on symmetric groups | |
Class 8 | evaluation of progress | |
Class 9 | Discussion session on homomorphisms of groups and image and kernel of a homomorphism of groups | |
Class 10 | Discussion session on normal subgroups and residue groups | |
Class 11 | Discussion session on the first, second and third fundamental theorems on group homomorphisms | |
Class 12 | Discussion session on subgroups generated by subsets | |
Class 13 | Discussion session on conjugacy of elements, conjugacy classes, and centralizers | |
Class 14 | Discussion session on the class equation and its applications | |
Class 15 | Discussion session on actions of groups |
None in particular
P.J. Cameron : Introduction to Algebra (second ed.), Oxford Univ. Press, 2008.
N. Jacobson : Basic Algebra I (second ed.), Dover，1985.
M. Artin : Algebra (second ed.), Addison-Wesley, 2011.
N. Herstein: Topics in algebra, John Wiley & Sons, 1975.
A. Weil: Number Theory for Beginners, Springer-Verlag, 1979.
Brief exam and oral presentation for exercise problems. Details will be announced during a lecture.
Students are supposed to have completed [Linear Algebra I / Recitation], [Linear Algebra II], [Linear Algebra Recitation II], [Introduction to Algebra I (ZUA.A201)] and [Exercises in Algebra A I (ZUA.A202)].
Students are strongly recommended to take ZUA.A203: Introduction to Algebra II (if not passed yet) at the same time.