Monthly Archives: October 2019

Solution: 2019-16 Groups with abundant quotients

Suppose a group \(G\) has a finite index subgroup that maps onto the free group of rank 2. Show that every countable group can be embedded in one of the quotient groups of \(G\).

The best solution was submitted by 하석민 (수리과학과 2017학번). Congratulations!

Here is his solution of problem 2019-16.

GD Star Rating
loading...

Solution: 2019-15 Singular matrix

Let \( A, B \) be \( n \times n \) Hermitian matrices. Find all positive integer \( n \) such that the following statement holds:

“If \( AB – BA \) is singular, then \( A \) and \( B \) have a common eigenvector.”

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-14.

A similar solution was submitted by 하석민 (수리과학과 2017학번, +3). Late solutions are not graded.

GD Star Rating
loading...

Solution: 2019-14 Residual finite groups

A group \(G\) is called residually finite if for any nontrivial element \(g\) of \(G\), there exists a finite group \(K\) and a surjective homomorphism \(\rho: G \to K\) such that \(\rho(g)\) is a nontrivial element of \(K\).

Suppose \(G\) is a finitely generated residually finite group. Show that any surjective homomorphism from \(G\) to itself is an isomorphism.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-14.

Other solutions were submitted by 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3).

GD Star Rating
loading...