Cyclic set: Difference between generator and unit

[smithnya]

Hello everyone,

I've just begun a lesson on cyclic sets, but I am having problems determining a few concepts. One question will ask me to find the generators and the units of a cyclic set Z₈. I have become confused and realized that I did not understand the difference between a generator and a unit in a cyclic set. Could someone explain such difference?

 

[Deveno]

perhaps you mean cyclic groups, or more generally a cyclic ring?

in Z₈ there are two operations: addition modulo 8, and multiplication modulo 8. now 8 is not prime, and one consequence of this is that Z₈ - {0} is not a group under multiplication.

nevertheless, certain elements of (Z₈)* do indeed have inverses:

(3)(3) = 9 = 1 (mod 8), so 3 is its own inverse in Z₈ (under multiplication mod 8).

the elements that possesses inverses under multiplication modulo 8 are called units, and form a group, the group of units of Z₈ (sometimes denoted U(8)).

one can show that U(8) is NOT cyclic, so it does not have a single generator (it takes at least two).

for certain integers n, U(n) IS cyclic, and one can speak of a generator, or "primitive element".

as far as the ADDITIVE group of Z₈, that is (of course) cyclic, and it just so happens that the generators of Z₈ as a cyclic group (under addition modulo 8) are also the units (elements of U(8)).

there is, in fact, this theorem:

k is a unit of Z_n iff gcd(k,n) = ?

(see if you can guess the answer).

 

[smithnya]

Is an element only a unit if it possesses a multiplicative inverse then?

 

[Deveno]

Is an element only a unit if it possesses a multiplicative inverse then?

in a ring, yes.

ok, the cyclic group Z₈ is {0,1,2,3,4,5,6,7}, where the group operation is addition modulo 8. technically, i should write [k] instead of k, since these are NOT integers, but equivalence classes of integers, but this abuse of notation is common-place.

as a(n additive) group the generators of Z₈ are 1,3,5 and 7 (these are the only elements of additive order 8).

but as a ring, Z₈ is not a domain, because it has zero-divisors: for example, 2 and 4 are zero-divisors, since (2)(4) = 0 (mod 8). this also means 2 and 4 cannot possibly have inverses, because if (for example again) 4 had an inverse a:

((2)(4))a = 0a
2(4a) = 0
2(1) = 0
2 = 0, a contradiction.

this happens precisely because 4 and 8 have a common divisor.

 

[blue_raver22]

I can provide a clear explanation of the difference between a generator and a unit in a cyclic set. A cyclic set is a mathematical structure where each element can be generated by repeatedly applying a single operation to a starting element. In this case, the operation is addition and the starting element is 0.

A generator in a cyclic set is an element that, when repeatedly added to itself, can generate all the other elements in the set. In other words, a generator is an element that, when combined with itself a certain number of times, can reach any other element in the set. In the example given, Z8, the generators would be 1, 2, 4, and 8, as each of these numbers can be added to itself to reach all the other elements in the set (e.g. 2+2=4, 4+4=8).

On the other hand, a unit in a cyclic set is an element that, when combined with any other element in the set, results in that same element. In the case of Z8, the only unit would be 0, as adding 0 to any other element in the set would result in that same element.

In summary, a generator in a cyclic set is an element that can generate all other elements in the set through repeated addition, while a unit is an element that does not change the value of any other element when added to it. I hope this explanation helps clarify the difference between a generator and a unit in a cyclic set.