Problem 2. Let G be a fixed group, and define a binary operationx~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 bewritten as [D] for some diagonal matrix D € GL2(R)?

# Explanation:## Step 1: Proving ∼ is an equivalence relation### To prove ∼ is an equivalence…

Answered: Problem 2. Let G be a fixed group, and…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 24E: Find two groups of order 6 that are not isomorphic.

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