Cayley Table for Multiplicative Group ( mathbb{Z}_{20}^* ) under Modulo 20 and Element Orders

The problem involves constructing the Cayley table for the multiplicative group ( mathbb{Z}_{20}^* ) under modulo 20 and determining the order of each element. This group consists of the integers less than 20 that are coprime with 20. The group ( mathbb{Z}_{20}^* ) is isomorphic to one of the abelian groups: ( mathbb{Z}_8 ), ( mathbb{Z}_4 times mathbb{Z}_2 ), or ( mathbb{Z}_2 times mathbb{Z}_2 times mathbb{Z}_2 ).

1. Constructing the Cayley Table

The group ( mathbb{Z}_{20}^* ) is a subset of ( { 1, 3, 7, 9, 11, 13, 17, 19 } ) as these are the integers less than 20 that are coprime with 20. The operation is multiplication modulo 20. Here is the Cayley table for the group ( mathbb{Z}_{20}^* ):

x times 1 3 7 9 11 13 17 19 1 1 3 7 9 13 17 19 3 3 9 1 7 19 11 17 7 7 1 9 3 17 13 19 9 9 7 3 1 17 13 19 11 11 19 13 17 9 3 1 13 13 17 19 11 3 7 9 17 17 11 19 3 9 1 7 19 19 17 13 13 11 7 1

The table is constructed by multiplying each element by each other element, and then taking the result modulo 20.

2. Finding Orders of Elements

The order of an element in the group is the smallest positive integer ( k ) such that ( a^k equiv 1 pmod{20} ).

Order of 3:

To find the order of 3, we multiply 3 by itself repeatedly until we get 1: 31 3 (mod 20) 32 9 (mod 20) 33 27 ≡ 7 (mod 20) 34 81 ≡ 1 (mod 20) Therefore, the order of 3 is 4. Order of 7:

A similar process shows the order of 7 is 4:7^1 7 (mod 20) 7^2 1 (mod 20) 7^3 9 (mod 20) 7^4 3 (mod 20) 7^5 1 (mod 20) Therefore, the order of 7 is 4. Order of 9:

Checking 9, we see: 9^1 9 (mod 20) 9^2 81 ≡ 1 (mod 20) Therefore, the order of 9 is 2. Order of 1:

The identity element 1 has order 1 as it is always 1.

For 11, 13, 17, and 19, the process is similar, and we can see:

Order of 11:

11^2 1 (mod 20) Therefore, the order of 11 is 2. Order of 13:

13^4 1 (mod 20) Therefore, the order of 13 is 4. Order of 17:

17^4 1 (mod 20) Therefore, the order of 17 is 4. Order of 19:

19^4 1 (mod 20) Therefore, the order of 19 is 4.

From the orders, we can see that the group ( mathbb{Z}_{20}^* ) is isomorphic to ( mathbb{Z}_4 times mathbb{Z}_2 ).

3. Subgroup Generated by an Element of Order 4

A subgroup generated by an element ( a ) of order 4 is the set ( langle a rangle { a^0, a^1, a^2, a^3 } ). For example, considering the element 13, which has order 4, the subgroup generated by 13 is:

( langle 13 rangle { 13^0, 13^1, 13^2, 13^3 } { 1, 13, 9, 7 } )

This subgroup has 4 elements, and it is a proper subgroup of ( mathbb{Z}_{20}^* ).