Question

Show that a nonzero subgroup of an additive abelian divisible group need not also be divisible.
Please, help

Answer 1

@Zarkon

Answer 2

Good. Now, let's get to work ^^

Answer 3

I am waiting :)

Answer 4

Define a divisible group :)

Answer 5

\[nA:=\{na : a \in A\}=A\]

Answer 6

Right.

Answer 7

for all \(n\in Z\)

Answer 8

Now, one of the simplest divisible groups, what is it? :D

Answer 9

Q

Answer 10

Yup. And it has a pretty well known subgroup that doesn't quite follow the same rules, what is it? :D

Answer 11

Z

Answer 12

Yup. Z is a subgroup of Q.

Now, tell me, for any integer n, is it true that
nZ = Z

?


;)

Answer 13

nope

Answer 14

because you say "for any integer n", but it is true if it turn to "there exists" :) that is n =1, right?

Answer 15

Yes, but that's not enough to show that Z is a divisible group. It has to be true for any integer, n
lol

Suppose n was 2, instead?

Then 2Z = the set of even integers, right? Most certainly not Z at all :P

Answer 16

Question: We will go to general argument, right? not a particular case of Q and Z, right?

Answer 17

What do you mean by that?

All the question wants is for you to show that the subgroup of a divisible group need not be divisible itself.

All that needs is an example :D

Answer 18

Oh.... :) my stupid brain just think of expand the problem to general one. I thought that must be true for ANY case.

Answer 19

But have to confirm: You mean we just give out 1 case only (as above) and it is enough?

Answer 20

I see whenever they say :"Show that....... is Not...." , most the case is just giving out a counterexample.
but when they say: "Prove that......." it means I have to prove it is true for any case.
Am I right?
I am sorry for those questions. Studying Math in English is a double challenge to me. :)

Answer 21

@Supreme-Kurt are you kicked out of the net again? hahaha.... It happens to me also.

Answer 22

I have class in 10 minutes, have to go. Thanks for respond @Supreme_Kurt. :)

Answer 23

you just need to show the above case

Answer 24

"show" and "prove" in some cases can be interchanged