Category Archives: problem

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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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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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

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

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2019-10 Is there canonical topology for topological groups?

Let \(G\) be a group. A topology on \(G\) is said to be a group topology if the map \(\mu: G \times G \to G\) defined by \(\mu(g, h) = g^{-1}h\) is continuous with respect to this topology where \(G \times G\) is equipped with the product topology. A group equipped with a group topology is called a topological group. When we have two topologies \(T_1, T_2\) on a set S, we write \(T_1 \leq T_2\) if \(T_2\) is finer than \(T_1\), which gives a partial order on the set of topologies on a given set. Prove or disprove the following statement: for a give group \(G\), there exists a unique minimal group topology on \(G\) (minimal with respect to the partial order we described above) so that \(G\) is a Hausdorff space?

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