If you're talking about group theory, you probably mean addition modulo n.
I have two questions about modulo n:
1). In my notes I have quoted everywhere! My big problem is: how do I know if this is addition modulo n or multiplication modulo n?
2). My second question is that I have this:
.
What does this mean? I know it's the direct sum, so does it mean this:
For example, ?
(I just went through and added the 1st number of each set to the 1st number of the other (and so on)). My big problem was that I didn't have anything to add to the 3 =S
If someone can clarify these it would help a lot!
is just a number system at this point, there is no operation. You can apply either addition or multiplication to it.
Not 100% sure here but this is what I would suggest.
I would take the first element of the first set and add it to each element of the second, then take the second element of the first set and add it to every element of the second and so on.
will be the set of distinct solutions.
Firstly. I assume you know what is? The cartestian product . Well it is fairly simple to show that if are groups then is a group with . When the two groups are abelian it is customary to write in lieu of
Secondly, you need to be a tad careful here. Firstly it should be . The equals sign is clearly false since the LHS is a set of ordered pairs and the RHS is not. But as is indicated the two groups are "isomorphic" (I believe from previous threads that you are familiar with this concept). To see why the above is true here is a brief outline. For to be cyclic we need there to be a generator of order . Note that for any it is true that (why?). Therefore, to have (and this of course is not a formal proof) a generator of order we must have that . So if then is a cyclic group of order . Also, you should know that any cyclic group of order is isomorphic to . From there the rest is trivial.