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
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | - | - |
Students are given exercise problems related to what is taught in the course "Introduction to Algebra II'''.
Course schedule | Required learning | |
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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 | Details will be announced during each lecture. |
Class 3 | Discussion session on the definition of a subgroup, criterion for subgroups, and examples of subgroups | Details will be announced during each lecture. |
Class 4 | Discussion session on right- and left-cosets by a subgroup | Details will be announced during each lecture. |
Class 5 | Discussion session on the order of a group and Lagrange's theorem | Details will be announced during each lecture. |
Class 6 | Discussion session on the order of an element of a group and on cyclic groups | Details will be announced during each lecture. |
Class 7 | Discussion session on symmetric groups | Details will be announced during each lecture. |
Class 8 | evaluation of progress | Details will be announced during each lecture. |
Class 9 | Discussion session on homomorphisms of groups and image and kernel of a homomorphism of groups | Details will be announced during each lecture. |
Class 10 | Discussion session on normal subgroups and residue groups | Details will be announced during each lecture. |
Class 11 | Discussion session on the first, second and third fundamental theorems on group homomorphisms | Details will be announced during each lecture. |
Class 12 | Discussion session on subgroups generated by subsets | Details will be announced during each lecture. |
Class 13 | Discussion session on conjugacy of elements, conjugacy classes, and centralizers | Details will be announced during each lecture. |
Class 14 | Discussion session on the class equation and its applications | Details will be announced during each lecture. |
Class 15 | Discussion session on actions of groups | Details will be announced during each lecture. |
Shoichi Nakajima : Basics of Algebra and Arithmetic, Kyoritsu Shuppan Co., Ltd., 2000.
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.