Unit 1 of 10
Abstract algebra
Six or seven questions every time, mostly short calculations on groups and fields.
- 6–7questions expected next time
- 7asked in 2025
- 6asked in 2026
- 2ideas asked in both papers
Know cyclic groups, p-groups and finite fields well. These come back in every paper.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Number of generators of a cyclic group of order n
Show answerHide answer
There are φ(n). For n = 8 that is 4.
Is R/I a field
Show answerHide answer
R/I is a field exactly when I is maximal. ℤₚ[x]/(f) is a field exactly when f is irreducible, and then it has p^(deg f) elements.
Asked in 2026
6 questions. Tap one for the answer and working.
Q36Is Z5[x]/(x^2+2) a field, and its orderA
Answer: field of order 25
x^2+2 has no root mod 5 (squares mod 5: 0,1,4; -2=3), so irreducible; Z5[x]/(x^2+2) is a field of 25 elements.
Sure
Q47Phi: Z[x] to Z6, f maps to f(1) mod 6: determine kernelB
Answer: 6, x-1
ker = {f : f(1) = 0 mod 6} = (6, x-1) (not principal).
Sure
Q52Count elements of F27 satisfying x^n=1 (exponent OCR-uncertain)D
Answer: 13
F27* cyclic of order 26 = 2*13; number of x with x^13=1 is gcd(13,26)=13.
Sure
Q56Group of order p^3 with center of order p: number of conjugacy classesLikely bonus marksB
Answer: p^2
|G|=p^3, |Z|=p: non-central classes have size p, count (p^3-p)/p = p^2-1, total p^2+p-1 (D4: 5). Not among options (p+2, p^2, p+1, p^3); p^2 is nearest.
Not sure
Misprint: the true count is p²+p−1 (5, 11, 29 for p = 2, 3, 5, checked by brute force). p² is the nearest option.
Q64G acts on itself by conjugation, all classes of size 1: G is abelian/trivial/simple/cyclicA
Answer: abelian
All conjugacy classes of size 1 iff G abelian (not necessarily trivial, cyclic or simple).
Sure
Q72Number of generators of cyclic group of order 105B
phi(105)=phi(3)phi(5)phi(7)=2*4*6=48 (checked)
Sure
Asked in 2025
7 questions. Tap one for the answer and working.
Q31Cyclic group of prime order is simple; Lagrange's theoremA
A true (prime order => only trivial subgroups => simple). R is Lagrange's theorem and is exactly why A holds.
Matches the official key
Q32HN/N ≅ ? (second isomorphism theorem)A
Second isomorphism theorem: HN/N ≅ H/(H∩N).
Matches the official key
Q33Number of generators of cyclic group of order 8C
Generators of C8 = phi(8) = 4.
Matches the official key
Q34Elements of A3D
A3 = {I, (1 2 3), (1 3 2)}.
Matches the official key
Q35Order of a given permutation in S4B
1->1, 2->3, 3->4, 4->2 is the 3-cycle (2 3 4); order 3.
Matches the official key
Q36Degree of splitting field of x⁴+x²+1 over QA
x^4+x^2+1 = (x^2+x+1)(x^2-x+1); all roots are primitive 3rd/6th roots of unity, in Q(w); degree 2.
Matches the official key
Q37M maximal iff R/M is a fieldD
M maximal iff R/M is a field (commutative ring with 1).
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Sylow theorems
- Cayley's theorem
- PID, UFD and ED distinctions
- Modules and Noetherian modules
- Hilbert basis theorem
Full syllabus for this unit
Groups, Cayley, homomorphism theorems, group actions, class equation, Sylow; rings, ideals, quotient rings, UFD/PID/ED, polynomial rings, fields, finite fields, extensions, modules, Noetherian modules, Hilbert basis theorem.