Is Zn a group under addition modulo n?

[roger]

Hi,

Firstly I am not sure what a group is, after being given the 4 group axioms.

I'm not sure what relevance ''closure'' has ?


and my question is : how to show that for every natural number,n, the set Zn of integers modulo n forms a group under addition modulo n ?

I appreciate any guidance.

Thankyou very much.

Roger

 

[HallsofIvy]

If you are going to look at groups as separate objects themselves, then "closure" under the group operation is essential. For example, if a and b are members of the group G but a+ b isn't then we are faced with the question of just what do we mean by "a+ b" if we are only talking about G!

To prove that any collection of things with a given operation is a group, you need to show that the definition of "group", usually given as a collection of "axioms", is satisfied.

To show that "Zn of integers modulo n forms a group under addition modulo n" you first have to decide what things are in that set (there are, in fact, a few different ways of looking at this). Assuming that you mean {0, 1, 2,..., n-1} then you must show:
I) Closure: that if m and n are both integers between 0 and n-1 then 0+ (n-1) (mod n) is also an integer between 0 and n-1.
II) Associativity: that (m+ n)+r= m+ (n+ r) (mod n)
(This is not just moving parentheses. The operation is defined on two objects at a time (m+n)+ r involves quite a different calculation than m+ (n+r).)
III) Identity: that there exist one member of the set, O, such that n+O= n for all n in the set.
IV) Inverse: that for any x in the set, there exist a y such that x+y= O.

 

[roger]

thanks Hallsoivy,

I don't actually know what the terms ''Zn of integers modulo n'' and '' addition modulo n" mean ?

and I am not sure how to find the elements of the set ?

and what is the difference in calculation for associativity exactly ?

I) Closure: that if m and n are both integers between 0 and n-1 then 0+ (n-1) (mod n) is also an integer between 0 and n-1.

What do you mean by this ?

 

[HallsofIvy]

thanks Hallsoivy,
I don't actually know what the terms ''Zn of integers modulo n'' and '' addition modulo n" mean ?
and I am not sure how to find the elements of the set ?
and what is the difference in calculation for associativity exactly ?
I) Closure: that if m and n are both integers between 0 and n-1 then 0+ (n-1) (mod n) is also an integer between 0 and n-1.
What do you mean by this ?

Then you do have a problem! The reason I asked about "deciding what things are in the set" is that there are a couple of different (but equivalent) ways of defining "Zn". The simplest is to take {0, 1, 2, ...,n-1} as the elements, then define "a+ b (mod) n" to be "the remainder when a+b is divided by n". For example, in Z₅, the elements are {0, 1, 2, 3, 4}
4+ 4 (mod) 5 = 3 because 4+4= 8 and 8= 5+ 3. Notice that, although 8 is greater than 5, "8 (mod 5)" is one of {0, 1, 2, 3, 4, 5}. That's what we mean by "closure". We don't go outside the set we are looking at.

It should be clear that 0 is the group identity in Z_n: n+ 0= n and since n is already less than n, you don't need to worry about the "mod" part.
What about "negatives" (additive inverse). If we were talking about Z₅, 3+ 2= 5 which has remainder 0 when divided by 5 so 3+ 2= 0 (mod 5). What "x" gives 4+ x= 0 (mod 5)? (In other words, what x gives 4+ x= 5?)
Can you generalize that to any Z_n?

 

[matt grime]

Associativity comes down to this:

suppose we have some way of composing two elements of something we want to test for "groupiness", then there are two ways one can bracket the a priori undefined object fgh as (fg)h or f(gh) are the same so that the symbol fgh is unambiguous.

Example: additoin we all know that doing (2+3)+4 is the same as doing 2+(3+4)

Counter example: subtraction is not associative. We know that the symbol a-b-c is not strictly well defined in the sense that

(1-1)-1

and

1-(1-1)

give different answers.

The best way to show something is associative is to appeal to the fact that the objects you're composing are derived from some other group where we know associativty holds. In the case of mod n arithmetic we can use the fact that addition in the integers is associative (which is true by fiat) to conclude it passes down to associativity for Z_n

Note the subscript: it is important. Zn is striclty different from Z_n in a fundamental way in the standard usages of these terms. Zn is usually the set of (integer) multiples of n inside Z. This too is a group.