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}.
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 , the pair of elements have same order and in have same order.
The unity in residue class group is .
Observe that are the inverses of each other in this group.
So, are inverses of each other, are inverses of each other.
In a group, an element's inverse order and position are equal.
,
Now, consider the multiplicative group
The identity element of is
So, and are inverses of each other
Similarly.
and are inverses of each other.
Since, order of element and its inverse is equal.
t follows that and
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