Abstract Algebra Course: What CUET PG, GATE and CSIR NET Need

An abstract algebra course should cover groups, rings and fields, but exams ask for different slices, and GATE and CSIR NET each name topics the other omits. CUET PG asks for the basics. GATE and CSIR NET share Sylow theorems and field extensions. GATE alone names group actions; CSIR NET alone names Galois theory.

Terms used in the table:

• A group is a set with one associative operation that has an identity and an inverse for every element, such as the integers mod 12 under addition.
• A ring has addition and multiplication, as polynomials do.
• A field is a ring with commutative multiplication in which every non-zero element has a multiplicative inverse, such as the integers mod 11.
• An ideal is a subset of a ring you can "divide out" by, the way multiples of 4 inside the integers give arithmetic mod 4.
• A homomorphism is a map between groups or rings that preserves their operations.
• Sylow theorems are counting rules for subgroups of prime-power order inside a finite group.
• A group action is a group acting by permuting the elements of a set.
• A finite abelian group is a finite group in which the order of the operation never matters.
• An irreducible polynomial cannot be written as a product of two polynomials of lower degree.
• A Euclidean domain is a ring with division with remainder, such as the integers. Every Euclidean domain is a principal ideal domain (PID), where every ideal is generated by one element. Every PID is a unique factorisation domain (UFD), where non-zero elements factor into irreducibles in essentially one way.
• Eisenstein's criterion is a test that proves certain integer polynomials irreducible.
• Cayley's theorem says every group is a group of permutations.
• The class equation counts the elements of a finite group by splitting it into conjugacy classes (elements that are the same up to relabelling by the group).
• Galois theory ties field extensions to groups of symmetries.

We built this table in October 2026 by reading the algebra paragraph of each syllabus line by line. The sources are the CSIR NET common syllabus, the GATE 2027 Mathematics syllabus (Section 6, Algebra) and the CUET PG Mathematics (SCQP19) syllabus. Linear algebra sits in a separate block of each syllabus and is not counted here.

The lists do not nest. A candidate preparing for GATE and CSIR NET together must cover the union of the two columns, not just the longer one. A CUET PG candidate who buys a course built around Galois theory pays for time the exam cannot use. For the full topic list of CSIR NET, see our CSIR NET Mathematical Sciences syllabus explained. For CUET PG, use our CUET PG syllabus guide.

In the official GATE 2026 Mathematics paper, 6 of 65 questions and 9 of 100 marks were abstract algebra. Most asked for a calculation or a true-or-false check on a given structure such as ℤ₃₀ ⊕ ℤ₁₂ or polynomials over ℤ₁₁. We counted them ourselves from the published question paper.

How we counted. In October 2026 we downloaded the official GATE 2026 Mathematics (MA) question paper, split its text at the question numbers (65 questions) and read each one. We called a question abstract algebra if it was about a group, ring, ideal, field or polynomial ring. Question 11 (a quotient of vector spaces) and the matrix questions went to linear algebra. Question 41 is a judgement call; without it the count is 5 questions and 7 marks. Marks follow the paper's own scheme: 1 mark for Q1 to Q5 and Q11 to Q35, 2 marks for Q6 to Q10 and Q36 to Q65.

Four of the six questions are about rings, polynomial rings and fields, and two are about finite groups. Q14 and Q41 test whether you recognise which theorem or definition applies. The other four need a calculation or a check on an explicit structure. Several are multi-select ("which of the following is/are TRUE"), so you must judge every option. One year is a sample of the pattern, not a quota.

These questions reward a theorem used as a tool. Take the group of order 595 = 5 × 7 × 17. The number of Sylow 5-subgroups must divide 119 and leave remainder 1 on division by 5. Of the divisors 1, 7, 17 and 119, only 1 works. So there is exactly one subgroup of order 5, it is normal, and the group always has a proper normal subgroup. That is a ninety-second argument if the theorem is fluent.

Answer these five questions without notes in ten minutes. Four or five correct means start timed previous-year practice. Two or three means revisit quotient groups and ideals first. Zero or one means work through groups with examples before any exam-focused course. The bands and the time limit are our recommendation, not an official benchmark. The questions are modelled on the question types above.

1. List all subgroups of ℤ₁₂ (integers mod 12 under addition).
2. Is ℤ₃₀ ⊕ ℤ₁₂ cyclic? What is the order of the element (14, 7)? (Here ⊕ means pairs (a, b) with a taken mod 30 and b mod 12, added coordinate by coordinate.)
3. Is x² + x + 4 irreducible over ℤ₁₁? Is ℤ₁₁[x]/⟨x² + x + 4⟩ a field?
4. On ℚ[x], is p(x) ↦ p(x²) a ring homomorphism? Is it onto?
5. A group has order 595. How many Sylow 17-subgroups can it have?

Answers

1. Six subgroups, one per divisor of 12. They are generated by 0, 6, 4, 3, 2 and 1, with orders 1, 2, 3, 4, 6 and 12.
2. Not cyclic, because gcd(30, 12) = 6, not 1. The order of 14 in ℤ₃₀ is 30/2 = 15 and the order of 7 in ℤ₁₂ is 12, so the order of (14, 7) is lcm(15, 12) = 60.
3. Irreducible. A quadratic over ℤ₁₁ has a root exactly when its discriminant (b² − 4ac) is a square mod 11. Here the discriminant is 1 − 16 = −15 ≡ 7 (mod 11), and the squares mod 11 are 0, 1, 3, 4, 5 and 9, so there is no root. Since ℤ₁₁[x] is a PID, an ideal generated by an irreducible polynomial is maximal, so the quotient is a field with 121 elements.
4. Yes, it is a ring homomorphism, because it preserves sums, products and 1. It is not onto, because x itself is never an output; only polynomials in even powers of x are.
5. Either 1 or 35. The number must divide 35 and be 1 mod 17, which allows 1 and 35 (35 = 2 × 17 + 1).

If algebra is where your mock tests lose marks, our CSIR NET algebra weakness diagnostic helps locate which topics are costing you.

Abstract algebra feels harder because the objects stop being numbers. In a quotient group the elements are whole sets, called cosets, and every operation has to give the same answer whichever member of a set you pick. Calculus gives procedures; this subject asks you to check definitions.

A common stumbling block is the quotient group, and one concrete example takes it down. Let G = ℤ₁₂ and H = {0, 4, 8}. A coset is a shifted copy of H, and there are four of them:

• 0 + H = {0, 4, 8}
• 1 + H = {1, 5, 9}
• 2 + H = {2, 6, 10}
• 3 + H = {3, 7, 11}

Each coset is one element of the quotient group, written ℤ₁₂/H. Add cosets by adding any representatives: (1 + H) + (3 + H) = 4 + H = H. Now check that the choice of representative does not matter, using 5 in place of 1 and 11 in place of 3: 5 + 11 = 16 ≡ 4 (mod 12), which lies in H, so the answer is the same coset. The four cosets behave exactly like ℤ₄.

Two points to hold on to:

• The order you list elements inside a coset is irrelevant; a coset is a set.
• A subgroup H is normal when gH = Hg for every g in the group, and normality is what guarantees the representative check always passes. In an abelian group such as ℤ₁₂ every subgroup is normal, so the example never fails. In the non-abelian group S₃, the subgroup {e, (12)} is not normal, and multiplying cosets by representatives gives conflicting answers. Try it once by hand.

Calculus is not a prerequisite. What you need is comfort with modular arithmetic, sets and writing short proofs.

Self-study or free NPTEL lectures work if you can finish a set of exercises each week without prompting. A paid course earns its fee when you need a fixed schedule, a person to answer doubts and previous-year questions solved in exam style. NPTEL (National Programme on Technology Enhanced Learning), the lecture programme run by the Indian Institutes of Technology, hosts Introduction to Abstract Group Theory from Chennai Mathematical Institute.

Whichever route you choose, run these five checks. We publish this page and we sell a course, so run the same checks on ours.

If you need only abstract algebra, our recorded Abstract Algebra course costs ₹1,999 as of October 2026 (listed at ₹2,999). Rishabh Sir teaches about 53 hours of group theory and ring theory in Hindi and English, with a doubt class, previous-year-question sessions and 430 pages of notes. It does not cover field extensions or Galois theory, which GATE and CSIR NET name.

For the whole syllabus, our CSIR NET and GATE Mathematics 2027 course teaches abstract algebra, alongside the other GATE and CSIR NET subject areas on the course page, including linear algebra, real analysis and complex analysis. Classes run live online with recordings, in Hindi and English. As of October 2026 the price is ₹15,999 one time, or four instalments of ₹3,999.75.

Vivek Sir teaches Abstract Algebra and Real Analysis. SB Sir (Sunil Bansal, who secured all-India rank 11 in CSIR NET) designs all our test series and conceptual tests, including the Core Concept Diagnostic test series. The course includes PDF notes, assignments, previous-year-question discussion, mock tests with analytics, and weekly doubt sessions or personal mentoring.

You can pass check 3 above before paying: we offer demo classes, free mathematics study content, PDF Drive resources and a ₹999 demo plan on the course. Our digital products carry a no-refund policy (a serious health issue immediately after purchase can be considered), so use a demo class first.

If you are preparing for CUET PG, read what recorded CUET PG mathematics classes should include before you buy anything, then ask us which of our recorded courses fits. Our support team answers 9am to 7pm, Monday to Saturday, and contact-form enquiries receive a reply within 24 hours.

To start, take the self-test above, then open the CSIR NET and GATE Mathematics 2027 course and try a demo class.