Intro Group Theory Problem:

Let G be a group.

(1) Define a relation on G by setting a b if and only if a = b ¹ . Use Problem 1 to show that this relation is symmetric.

(2) Show that if G has an even number of elements, then there is an element a not = e such that a = a ¹ .

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Problem is in the attached picture

Let G be a group [1) Dene a retation on G by setting a m tr if and only ifa = Err1. Use Prahtem 1 to show that this relation is symmetric. [2) Show that if G has an even nnmher of elements, then there is an element a 5-5 e such that a = a'l. (Hint: eansitter the partition of G assaeiatett to the relation dened in the rst part of this prohtem {you might want to yo hast: and tent: at Thm 2.3.1). How many elements can there he in each equivalence etass 1?} [3) ed-nee from the point above that if G has an even number of ete- rnents, then there is an a E G, with a 3'5 e sneh that a2 = e. MDT E: Problem 1: Let G be a group. Show that for every element a E G, [a 1}I 1 = a. Theorem 2.3.1: Let I he an_].-r set. {i} Suppose that {Ii : i E i j is a partition th. Then the reta tion R on X which is defined by xii? if and onr if): and y hetongI to the same member of the partition is an equivatenee reia tion. {ii} It convex-set}; E is an equivatenee retation on I then E determines the partition whose 'hioeks' Xi are the equivatence etasses 1'st = r EX: yEx} of members it of X