d Prove whether or not there exists an element of order 11 i

d) Prove whether or not there exists an element of order 11 in Z100.

Solution

Corollory Of Lagrange \' s theorem in Group theory states

G is a group and any element a in G   

then O (a) divides O (G)

1. O (Z₁₀₀) = 100 if we take the group as (Z₁₀₀ , +₁₀₀)= { 0,1,2,----99}

then 11 doesnt divide 100 ten there is no element wose order is 11

2 . If the group is ( Z_{100 ,} X₁₀₀) = { 1,2,--99} where O (Z₁₀₀₎=99 as we delete 0

O(1) =1 here 1 is the identity element for X