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Step 1
a. Determine U 14, the set of units in the ring.
In the ring Z , the set of units U consists of all elements that have a
14 14
multiplicative inverse. An element a ∈ Z has a multiplicative inverse if and 14
only if a is relatively prime to 14, i.e., gcd (a, 14) = 1.
The numbers less than 14 that are relatively prime to 14 are 1, 3, 5, 9, 11, and
13. Thus, U = {1, 3, 5, 9, 11, 13}.
14
Step 2
b. Construct the multiplication table modulo 14 for U 14.
The multiplication table for U can be constructed by calculating the
14
products of the elements in U and then taking the result modulo 14.
14
Here is the multiplication table:
| × | 1 | 3 | 5 | 9 | 11 | 13 |
|--------|---| ---|---|----|----|---|
|1 | 1 | 3 | 5 | 9 | 11 | 13 |
|3 | 3 | 9 | 1 | 13 | 5 | 11 |
|5 | 5 | 1 | 11 | 3 | 13 | 9 |
|9 | 9 | 13 | 3 | 1 | 9 | 5 |
| 11 | 11 | 5 | 13 | 9 | 3 | 1 |
| 13 | 13 | 11 | 9 | 5 | 1 | 3 |
In this table, each cell represents the product of the row and column headers
modulo 14.
Step 3
c. To what group is ⟨U 14, ⋅⟩ isomorphic to?
The group ⟨U , ⋅⟩ is isomorphic to a group that has the same structure. In
14
this case, we can identify that U is a cyclic group because it has an element
14
of order 6 (the size of U itself), which is 3, since 3 ≡ 1mod14 and no
14
6
smaller power of 3 is congruent to 1 modulo 14.
Hence, ⟨U , ⋅⟩ is isomorphic to the cyclic group Z , the integers modulo 6
14 6
under addition, because both groups have 6 elements and are cyclic. The
isomorphism can be explicitly constructed by mapping the generator 3 of U 14
to the generator 1 of Z . 6
Answer
a. U 14 = {1, 3, 5, 9, 11, 13}
b. The multiplication table modulo 14 for U 14 is provided.
c. ⟨U 14, ⋅⟩ is isomorphic to Z 6.
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Wed Jan 17 2024 22:36:05 GMT+0800 (China Standard Time)
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