(a) Let H and K be two subgroups of a group G such that o(H) √o(G) and o(K) √o(G). Show that H ∩ K ≠ {e}, where e is the identity element. Here o(H), o(K) and o(G) denote the order of H, K and G respectively. (10 marks)

(b) Let G = {e, x, x², y,

Question

(a) Let H and K be two subgroups of a group G such that o(H) > √o(G) and o(K) > √o(G). Show that H ∩ K ≠ {e}, where e is the identity element. Here o(H), o(K) and o(G) denote the order of H, K and G respectively. (10 marks) (b) Let G = {e, x, x², y, yx, yx²} be a non-Abelian group with o(x) = 3 and o(y) = 2. Show that xy = yx² (where e is the identity element of G and o(x), o(y) denote the order of the elements x, y respectively). (10 marks) (c) Examine whether the series Σₙ₌₁^∞ (-1)ⁿ⁻¹/n is absolutely or conditionally convergent. (10 marks) (d) Expand f(z) = 1/(z+1)(z+3) in a Laurent series valid for 1 < |z| < 3. (10 marks) (e) How many basic solutions are there for the following system of equations? 2x₁ - x₂ + 3x₃ + x₄ = 6 4x₁ - 2x₂ - x₃ + 2x₄ = 10 Find all of them. Furthermore, find the number of basic solutions, which are feasible/non-feasible/non-degenerate. (10 marks)

UPSC Answer Check · Accepted Answer

This multi-part question requires proving results in group theory, analyzing convergence, expanding complex functions, and solving linear programming systems. Allocate approximately 15-18 minutes each for parts (a), (b), and (d) which demand rigorous proofs and careful Laurent series construction; spend 10-12 minutes each on (c) and (e). Begin each part with clear statement of what is to be shown, present logical derivation with theorems cited, and conclude with explicit verification of the required result.
- For (a): Apply the product formula |HK| = |H||K|/|H∩K| and use the bound |HK| ≤ |G| to derive contradiction if H∩K = {e}
- For (b): Use the non-Abelian property and orders of x, y to eliminate xy = yx and xy = yx², showing only xy = yx² satisfies group axioms
- For (c): Show Σ1/n diverges (harmonic series) for absolute convergence, then apply Leibniz test for conditional convergence
- For (d): Perform partial fraction decomposition, then expand 1/(z+1) as geometric series in 1/z for |z|>1 and 1/(z+3) as series in z/3 for |z|<3
- For (e): Identify m=2 equations, n=4 variables, so C(4,2)=6 basic solutions; classify each by checking non-negativity and degeneracy
- Correct enumeration of all six basic solutions with proper identification of basic/non-basic variables for part (e)

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	Correctly identifies all hypotheses: for (a) applies \|HK\| formula with proper inequality; for (b) uses orders to constrain products; for (c) sets up absolute vs conditional test; for (d) identifies annulus 1<\|z\|<3 correctly; for (e) determines C(4,2)=6 basic solutions	Most setups correct but misses one constraint (e.g., wrong annulus in (d), or miscalculates number of basic solutions in (e))	Fundamental setup errors: wrong formula for \|HK\|, incorrect region identification, or confuses basic with feasible solutions
Method choice	20%	10	Selects optimal methods: Lagrange's theorem/contradiction for (a), elimination using group properties for (b), Leibniz test for (c), partial fractions with appropriate geometric series expansions for (d), systematic enumeration for (e)	Correct methods but suboptimal (e.g., direct expansion without partial fractions in (d), or missing efficient counting in (e))	Inappropriate methods (e.g., ratio test alone for conditional convergence, or solving by substitution rather than basis enumeration)
Computation accuracy	20%	10	All calculations precise: inequality manipulation in (a), product verification in (b), series summation bounds in (c), coefficient extraction in Laurent series, exact solution values in (e)	Minor computational slips (e.g., arithmetic error in one basic solution, or sign error in series expansion)	Major computational errors: incorrect inequality direction, wrong coefficients in series, or multiple incorrect basic solutions
Step justification	20%	10	Every step justified with named theorems: Lagrange's theorem, product formula, Leibniz test conditions, geometric series validity conditions, linear programming definitions; logical flow explicit	Most steps justified but gaps in reasoning (e.g., assumes convergence without verification, or skips degeneracy check)	Unjustified leaps, missing theorem citations, or circular reasoning; asserts results without proof
Final answer & units	20%	10	Clear final statements: explicit contradiction established in (a), xy=yx² verified in (b), 'conditionally convergent' stated in (c), correct Laurent series with region specified in (d), complete table of 6 solutions with feasibility/degeneracy classification in (e)	Correct answers but poorly presented (e.g., missing region in (d), incomplete classification in (e))	Missing or wrong conclusions; fails to answer what was asked; no classification of solutions in (e)

Q1

Directive word: Prove

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Mathematics 2025 Paper II