# 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.

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# Solution: 2019-12 Groups generated by two homeomorphisms of the real line

Let $$I, J$$ be connected open intervals such that $$I \cap J$$ is a nonempty proper sub-interval of both $$I$$ and$$J$$. For instance, $$I = (0, 2)$$ and $$J = (1, 3)$$ form an example.

Let $$f$$ ($$g$$, resp.) be an orientation-preserving homeomorphism of the real line $$\mathbb{R}$$ such that the set of points of $$\mathbb{R}$$ which are not fixed by $$f$$ ($$g$$, resp.) is precisely $$I$$ ($$J$$, resp.).

Show that for large enough integer $$n$$, the group generated by $$f^n, g^n$$ is isomorphic to the group with the following presentation

$<a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>.$

The best solution was submitted by 김동률 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-12.

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# Notice on POW 2019-12

POW 2019-12 is still open and anyone who first submits a correct solution will get the full credit.

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# Solution: 2019-13 Property R

Let $$A_{a, b} = \{ (x, y) \in \mathbb{Z}^2 : 1 \leq x \leq a, 1 \leq y \leq b \}$$. Consider the following property, which we call Property R:

“If each of the points in $$A$$ is colored red, blue, or yellow, then there is a rectangle whose sides are parallel to the axes and vertices have the same color.”

Find the maximum of $$|A_{a, b}|$$ such that $$A_{a, b}$$ has Property R but $$A_{a-1, b}$$ and $$A_{a, b-1}$$ do not.

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

Here is his solution of problem 2019-13.

An incorrect solution was received. Late solutions are not graded.

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# 2019-13 Property R

Let $$A_{a, b} = \{ (x, y) \in \mathbb{Z}^2 : 1 \leq x \leq a, 1 \leq y \leq b \}$$. Consider the following property, which we call Property R:

“If each of the points in $$A$$ is colored red, blue, or yellow, then there is a rectangle whose sides are parallel to the axes and vertices have the same color.”

Find the maximum of $$|A_{a, b}|$$ such that $$A_{a, b}$$ has Property R but $$A_{a-1, b}$$ and $$A_{a, b-1}$$ do not.

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# Notices

1. There will be no POW this week due to 추석 (thanksgiving) break. POW will resume next week.

2. The submission due for POW2019-12 is extended to Sep. 18 (Wed.).

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Let $$I, J$$ be connected open intervals such that $$I \cap J$$ is a nonempty proper sub-interval of both $$I$$ and$$J$$. For instance, $$I = (0, 2)$$ and $$J = (1, 3)$$ form an example.
Let $$f$$ ($$g$$, resp.) be an orientation-preserving homeomorphism of the real line $$\mathbb{R}$$ such that the set of points of $$\mathbb{R}$$ which are not fixed by $$f$$ ($$g$$, resp.) is precisely $$I$$ ($$J$$, resp.).
Show that for large enough integer $$n$$, the group generated by $$f^n, g^n$$ is isomorphic to the group with the following presentation
$<a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>.$