Proof Using Def. of Groups and Def. of Subgroups

[wubie]

Hello,

This is my question:

(i) Let H and K be subgroups of a group G. Prove that the intersection of H and K is also a subgroup of G.

(ii) Give an example, using suitable sugroups of the goup of integers with the operation addition, (Z,+), to show that if H and K are subgroups of a group G, then the union of U and K need not be a subgroup of G.


I figure that if I can do (i) then (ii) will follow. But I am unsure of how to do (i). (In previous assignments I did proofs on subsets, unions, intersections of sets but I did poorly on them.)

I posted a previous question stating the four properties of a set of which all must be satisfied to be defined as group. From my understanding of subgroups, only two of these properties must be satisfied to define a subgroup of a group.

Definition: A subgroup of a group G (G,o) is any nonempty subset H of G such that H is a group with the same operation, o. To check that H is a subgroup, verify the following:

S1: If x and y are elements of H, then x o y is an element of H.
s2: If x is an element of H, the the inverse is an element of H.


If H and K are subgroups then they must satisfy the aforementioned properties. And as a consequence if S1 and S2 are satisfied, and because H and K are elements of G, the remaining 2 properties that define a group are satisfied as well.

I also know that by definition, H intersects K iff some of the elements of K are also elements of H.

But I am unsure of where to go from here, much less construct a proof that proves that the intersection of H and K is also a subgroup of G.

(After typing this I think that I have some idea so I might be back later to submit more ideas/work regarding this question).

Any help to steer me in the right direction would be appreciated. Thankyou.

 

[quantumdude]

Your problem is Proposition 2.1.2 at the following website. The complete proof is there, so you may want to only read a line at a time for a clue.

http://www.maths.lancs.ac.uk/dept/coursenotes/m225ril99/chapter2/chap2/node2.html

 

[wubie]

Thanks Tom. I might get back to you if I don't understand a step or few. Or for some clarification of some concept. That happens from time to time you know.

Thanks again.

 

[HallsofIvy]

I also know that by definition, H intersects K iff some of the elements of K are also elements of H.

But it's not just a matter of WHETHER they intersect but WHAT the intersection is.

To check that H is a subgroup, verify the following:

S1: If x and y are elements of H, then x o y is an element of H.
s2: If x is an element of H, the the inverse is an element of H.

Yes, exactly. Suppose x and y are members of H intersect K.
Then x and y are members of ____ and, since ____ is a subgroup, x+y is in ____.

Since x and y are members of H intersect K, they also are members of ___ which is a subgroup. Therefore, x+y is a member of _____.

Since x+y is in ___ and ___, it is in ___ intersect ___.

Fill in the blanks!