List the units in each of the following: Z_T- Z_6 Z_8 Draw up the table for mult

Question

List the units in each of the following: Z_T- Z_6 Z_8 Draw up the table for multiplication (mod 8) in U(8). Is this a group? Explain why or why not. Determine the inverse of 3 in U(8). Draw up the table for multiplication (mod 10) in U(10)., Is this a group? If not explain why not.Is there any positive integer n for which U(n) with multiplication (mod n) will fail to be a group? Either provide an for which U(n) is not a group, or explain why U(n) is a group for all positive integer s n Make a conjecture: What condition on n ensures that the underlying set for U(n) is precisely Z_n^+? Make a conjecture: for a given positive integer n, how can one identify the units in Z_n without writing out the whole table for multiplication mod n on Z_n?

Explanation / Answer

1) List the units in Z7, Z6 and Z8

In Z7 all the non-zero elements are units as can be seen below:

              1.1 =1, 2.4=1, 3.5=1, 6.6 =1

In Z6 the units are 1 and 5 (the other elements are zero divisors, 2.3 =0, 4.3=0)

            1.1=1 and 5.5 =1

In Z8 the units are 1,3,5 and 7 (the other elements are zero divisors--2.4=0, 6.4=0)

           3.3=1, 5.5 =1 and 7.7=1

2) As seen above U(8) = {1,3,5,7} and the multiplication table is given by

From the table, it is seen that U(8) is closed under multiplication, 1 belongs to it and every element has an inverse. So U(8) is a group.

3) Multiplication table for U(10) ={1,3,7,9}

U(10) is a group (as can be seen from the table, it is closed under multiplication, identity exists and inverse exists for every element )

4) Every U(n) is a group indeed. U(n) by definition is the set of all invertible elements in the ring Zn , and hence is a group.

5) U(n) (for the examples seen above) is the set {x: x in Zn - {0} , x is relatively prime to n}.

So we expect U(n) to be the multiplicative group Znx if n is a prime .(as in this case , all non-zero x in Zn are relatively prime to n.

6) As seen from the above, U(n) is conjectured to be the set {x: x in Zn - {0} , x is relatively prime to n}.

1 3 5 7 1 1 3 5 7 3 3 1 7 5 5 5 7 1 3 7 7 5 3 1

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