Without actually computing the orders, explain why the

Answered question

2022-03-08

Without actually computing the orders, explain why the two elements in each of the following pairs of elements from Z30 must have the same order: {2, 28}, {8, 22}. Do the same for the following pairs of elements from U(15): {2, 8}, {7, 13}.
 

Answer & Explanation

Ian Adams

Ian Adams

Skilled2022-03-23Added 163 answers

A group's elements and their inverses are arranged in the same order. The objective is to show that in Z30, the pair of elements {2,28},{8,22} have same order and in U(15),12,8,7,13 have same order.

The unity in residue class group Z30 is 0.
Observe that r,30-r are the inverses of each other in this group.
So, 2, 28 are inverses of each other, 8, 22 are inverses of each other.
In a group, an element's inverse order and position are equal.
|2|=|28|
|8|=|22|

|2|=|28||8|=|22|

Now, consider the multiplicative group U(15)={1,2,4,7,8,11,13,14}

The identity element of U(15) is 1 

2×8=16 

=1(modulo 15) 

So, 2 and 8 are inverses of each other 

Similarly. 

7×13=91 

=1(modulo 15) 

7 and 13 are inverses of each other. 

Since, order of element and its inverse is equal. 

t follows that |2|=|8| and |7|=|13|

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