2016 Exercises in Algebra A II

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Taguchi Yuichiro  Kawachi Takeshi  Yamakawa Daisuke 
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Course description and aims

This course is an exercise session for "Introduction to Algebra II'' (ZUA.A203). The materials for exercise are chosen from that course.

Student learning outcomes

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

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Students are given exercise problems related to what is taught in the course "Introduction to Algebra II'''.

Course schedule/Required learning

  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

Reference books, course materials, etc.

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.

Assessment criteria and methods

Brief exam and oral presentation for exercise problems. Details will be announced during a lecture.

Related courses

  • MTH.A203 : Introduction to Algebra III
  • MTH.A204 : Introduction to Algebra IV
  • ZUA.A201 : Introduction to Algebra I
  • ZUA.A202 : Exercises in Algebra A I
  • ZUA.A203 : Introduction to Algebra II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

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.

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