group theory

ANORRYNFYNZ
timer Asked: Jan 4th, 2022

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STUDENT NAME; Muhammad Akhtar STUDENT ID ; MC210401320 Q1: Find the order of each element of the group 1, − 1, i, − i Solution : Let G= 1, − 1, i, − i Because this group under multiplication so e = 1 By using Formula a n =e  a  G Here n is called the order of the element Putting a = 1 and n = 1 in the formula (1) 2 = 1 = e , So, 1 is order of 1 Now, Putting a = − 1 and n = 2 in the formula ( -1) 2 = 1 = e So, 2 is order of − 1 Similarly, Putting a = i and n = 4 in the formula ( -1) 4 = 1 = e So, 4 is order of i And , Putting a = −i and n = 4 in the formula ( -1) 4 = 1 = e So, 4 is order of − i Q 2 : Show that the set of integers ‹Z, +› is a group under addition. Solution : Case 1: There exists an identity element in the group that fixes every element under given the binary operation. Yes, the number 0 is the identity since 0 + k = k + 0 k Z Case 2 : Closure : given any two _ element a and b in the set , we need to show that a + b in the set. Yes, given any two integers a and b, there same a + b is again an integer. Case 3 : Associativity : Yes, addition in Z is associative. i.e., ( a+b ) c= a ( b+c ) Case 4 : Existent of inverse : given any element a in the set , we need to find another element such that a and its inverse commute and their operation together given the identity element. Yes, given a  G then − a is the inverse. Since a+(-a)=(-a)+a=0 Since Z with addition satisfies above all 4 conditions. So,‹Z, +› is a group under addition
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