# Structures - GROUPS

• May 19th 2013, 02:26 PM
YoungStudent
Structures - GROUPS
Hy guys .... I hope that are all fine with you ....

I had this exercises to do, and I'm having problems to solving them, can some of you,help me?

**** find the subgroup of the group (Z, +, 0) generated by the set {10, 15} .

best Regards
• May 19th 2013, 10:58 PM
chiro
Re: Structures - GROUPS
Hey YoungStudent.

First consider all possible situations of 10x + 15y where x and y are whole numbers (in Z) and then consider that you need to find a minimum set that satisfies this criteria.

Hint: Consider multiples of 5 and see if you can construct every multiple of 5 with the linear combination 10x + 15y.
• May 20th 2013, 05:28 AM
YoungStudent
Re: Structures - GROUPS
Hi there.. thank you very much for your time and explanation.

So, in your hint, you talk about 5, and I presume that you get it, from here right: (-1)*10 + 1*15) =5, that will be our smallest positive integer in the subgroup ?
• May 20th 2013, 05:56 AM
HallsofIvy
Re: Structures - GROUPS
"{Z, +, 0}" is the group of all integers with "+" as operation and "0" as the additive identity. {10, 15} "generate" the subgroup of all integers we can get by starting with 10, 15, -10, and -15 and adding them any number of times. If we call the number of times we add 10 n and the number of times we add 15 m, we get 10n+ 15m as a formula for all such numbers. In particular, we can write this as 5(2n+ 3m) showing that all numbers in this set are multiples of 5. What about the other way? If we can show that all multiples of 5 as in this set, we are done.
Suppose 10n+ 15m= 5k for some integer k. We can immediately divide by 5 to get 2n+ 3m= k.

Start by looking at 2n+ 3m= 1. It is obviously true that m= 1, n= -1 is a solution. Multipying by k gives m= k, n= -k such that 2n+ 3m= 2(-k)+ 3(k)= k. That is, given any integer k, there exist integers n and m such that 2n+ 3m= k and so that 10n+ 15m= 5k.

The subgroup generated by 10 and 15 is the subgroup of all multiples of 5.