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College Mathematics Corner

Groups with no subgroups

Claim: If G has no subgroups H /= (e), G, then G must be cyclic of prime order. Proof:  One …

Trace of product of matrix (TIFR 2015 problem 3)

Question: Let \(A\) be a \(10\times 10\) matrix with complex entries such that all its eigenvalues are non-negative real numbers, …

Image of continuous function (TIFR 2015 problem 2)

Question: Let \(f: \mathbb{R} \to \mathbb{R} \) be a continuous function. Which of the following can not be the image of …

Invertible matrix with sum of each row 1 (TIFR 2015 problem 1)

Question: Let \(A\) be an invertible \(10 \times 10\) matrix with real entries such that the sum of each row is …

Homomorphisms from \(S_n\) (TIFR 2014 problem 23)

Question: There exists an onto group homomorphism A. from \(S_5\) to \(S_4\) B. from \(S_4\) to \(S_2\) C. from \(S_5\) …

Number of maps (TIFR 2014 problem 30)

Question: How many maps \(\phi: \mathbb{N} \cup  \{0\} \to \mathbb{N} \cup  \{0\}\) are there satisfying \(\phi(ab)=\phi(a)+\phi(b)\) , for all \(a,b\in …

Maps from compact spaces (TIFR 2014 problem 29)

Question: Let \(f: X\to Y \) be a continuous map between metric spaces. Then \(f(X)\) is a complete subset of …

Continuous functions from discrete space (TIFR 2014 problem 28)

Question: Let \(X\) be a topological space such that every function \(f: X \to \mathbb{R}\) is continuous. Then A. \(X\) …

Homeomorphism of product space with mother space (TIFR 2014 problem 27)

Question: \(X\) is a topological space with infinite cardinality which is homeomorphic to \(X \times X\). Then A. \(X\) is …

Number of irreducible polynomials (TIFR 2014 problem 26)

Question: The number of irreducible polynomials of the form \(x^2+ax+b\) , with \(a,b\) in the field \(\mathbb{F}_7\) of 7 elements …

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