Problem 2. Let G be a fixed group, and define a binary operation x~y if y = gxg‍1 for some g≤ G. 2.1. Show that ~ defines an equivalence relation on G. 2.2. Show that if two elements x, y Є G are related by y = ~ on G by = gxg1 for some gЄ G, then y = gang1 holds for all n € Z+. Conclude that if x ~y, then |x| = |y|. 2.3. Now let G = GL2(R). Is it true that every equivalence class of ~ can be written as [D] for some diagonal matrix D € GL2(R)?

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