I need an example of a union of 2 subgroups that isnt a subgroup

Question

swissgirl · Answer

It sounds more confusing than it is
Basically H and K are subgroups of G but $ H \cup K $ isnt a subgroup of G

anonymous · Answer

what groups do you know?

swissgirl · Answer

I know nothing

swissgirl · Answer

Cyclic groups

swissgirl · Answer

and that is it

anonymous · Answer

if you pick two subgroups of $S_3$ and take there union it will not be a subgroup

anonymous · Answer

but that one is not cyclic
how about the group $\mathbb{Z}$ under addition

swissgirl · Answer

ohhhhh It doesnt have to be cyclic

anonymous · Answer

subgroups of $\{\mathbb{Z},+\} $ look like what?

anonymous · Answer

Yeah, I can't think how this is even possible...

swissgirl · Answer

it is. I am in middle of working on $ (\mathbb{Z_4}, +) $

anonymous · Answer

yeah.  I am really weak on Set Theory.  Next thing to brush up on...

swissgirl · Answer

I am weak on everything -_-

anonymous · Answer

subgroups of $\{\mathbb{Z},+\}$ look like $n\mathbb{Z}$

anonymous · Answer

$$
A= 5 Z\
B=7 Z\
5 \in A \cup B\
7 \in A \cup B\
but\
5+7 \notin  A \cup B\

$$

swissgirl · Answer

huh I had a totally diff example

swissgirl · Answer

I dont get it what u guys r doing

anonymous · Answer

so take for example the union of two such subgroups, say $7\mathbb{Z}$ and $5\mathbb{Z}$

anonymous · Answer

oh wow i am being a copy cat, 
i guess it was the easiest answer

swissgirl · Answer

lol

anonymous · Answer

is it clear what i wrote (after @eliassaab  wrote the same thing first?)

swissgirl · Answer

Like i dont get ur notation.
They dont show any of that in my book

swissgirl · Answer

But I have another exxample

anonymous · Answer

as subgroup of $\mathbb{Z}$ under addition looks like all multiples of some number say 5
so it would be the set $\{...-15,-10,-5,0,5,10,15,...\}$

swissgirl · Answer

okk

anonymous · Answer

you can check that this is a subgroup easily
now pick another such subgroup
because apparently i have no originality pick 
$$7\mathbb{Z}=\{...-21,-14,-7,0,7,14,21,...\}$$

anonymous · Answer

their union is what? all multiples of 5 or 7
but it is not even closed, since $5+7$ is not in there

swissgirl · Answer

ohhhhh i get ittttttt

swissgirl · Answer

Thanksss idk all i see in my book are these mod examples so i just used one of those

swissgirl · Answer

Thankkkssssss guys