Group Theory

Passport

• Where and when. Semester III of program 01.03.02 “Applied Mathematics and Informatics”, profile “Mathematics”, in the curriculum for the 2025 intake.
• Workload. 135 hours, 3 credits: 60 contact hours (30 lectures and 30 seminars), 45 hours of independent work and 30 hours for the exam.
• Department. In the published exam program, the Department of Higher Mathematics; an adjacent extended course in year 2 is taught by the Department of Discrete Mathematics.
• Lecturer in the published 2025/26 program. V. V. Shtepin.
• Final assessment. Exam in semester III.

Contents of the published program

Below are all 24 items of V. V. Shtepin’s exam program for 2025. It belongs specifically to this course; sections on rings and fields from adjacent materials are not carried over here.

1. Groups, examples, cyclic groups and their subgroups.
2. Cosets, the index of a subgroup and Lagrange’s theorem.
3. Normal subgroups, the quotient group, homomorphisms, kernel and image; the fundamental homomorphism theorem.
4. The product of a normal subgroup and a subgroup, and the first isomorphism theorem.
5. Subgroups of a quotient group and the correspondence theorem.
6. Group actions, orbits, stabilizers and the orbit formula.
7. Examples of actions, Cayley’s theorem, the centralizer of an element and the normalizer of a subgroup.
8. Automorphisms and the group of inner automorphisms \(\operatorname{Inn}(G)\).
9. The class equation and Burnside’s lemma on the average number of fixed elements.
10. External and internal direct products, a criterion of decomposability into a direct product.
11. The semidirect product and a criterion for the corresponding decomposition.
12. The center of a group, the quotient by the center, and the center of a \(p\)-group.
13. Solvable and nilpotent groups, equivalent definitions, and relations with subgroups and quotient groups.
14. Simple groups and a lemma on conjugacy classes in a subgroup of index 2.
15. Simplicity of the group \(A_n\) for \(n\geq5\).
16. The free group, its quotient groups, and presentation of a group by generators and defining relations.
17. Existence in a \(p\)-group of subgroups of all admissible orders.
18. Existence and number of Sylow subgroups: the first and third Sylow theorems.
19. Conjugacy of Sylow subgroups and embedding of a \(p\)-subgroup into a Sylow subgroup: the second Sylow theorem.
20. Decomposition of a finite group into a direct product of Sylow subgroups; decomposition of a group of order \(pq\).
21. Free abelian groups, bases, rank, subgroups of free abelian groups.
22. The Smith normal form and existence of a decomposition of a finitely generated abelian group.
23. The torsion part of an abelian group and uniqueness of its classification decomposition.
24. Rings and algebras, ideals, the quotient ring, and the fundamental homomorphism theorem for rings.

Assessment

The open archive contains a grading scheme sheet for the course “Group Theory”, year 2, semester III, for 100 points: two tests on handing in the assignments, 0–8 each; two assignments with a notebook and a defense, 0–4 each; theory check, 0–3; seminar work, 0–3; oral exam, 0–70. For a preliminary oral grade \(N>2\), \(7N\) points are awarded; for \(N=1,2\) the final grade equals \(N\).

Scale: 93–100 — 10, 86–92 — 9, 79–85 — 8, 72–78 — 7, 65–71 — 6, 58–64 — 5, 51–57 — 4, 44–50 — 3, 30–43 — 2, 0–29 — 1. The sheet itself does not name V. V. Shtepin’s stream and was published before the current programs, so this is archival reference information, not a guarantee of the grading scheme in force.

Sources and currency

Shtepin’s exam program, 2025, archival grading scheme, extended program of the Department of Discrete Mathematics, curriculum 01.03.02.