1. ## Help:--Subgroups

Hello...
I need real help in these problems...

Q1) Prove that if H and K are subgroups of a group G (with operation *), then H intersection K is a subgroup of G.

Q2) Let H = {(1), (1 2)} and K = {(1), (1 2 3), (1 3 2)}. Both H and K are subgroup of S3. Show that HUK is not a subgroup of S3. It follows that a union of subgroup is not necessarily a subgroup.

2. Do you know what property of a set must be shown in order to prove the set is a subgroup? Please post what you know about this.

3. Sir, this is what I could find in the book:

Let G be a group with operation * and let H be a subset of G. Then H is a subgroup of G if and only if:
1) H is non empty.

2) If a E H and b E H then a*b E H

3) If a E H then a(inverse) E H.

4. Originally Posted by Vedicmaths
Hello...
I need real help in these problems...

Q1) Prove that if H and K are subgroups of a group G (with operation *), then H intersection K is a subgroup of G.
you were right in what you said you needed to check if a group is a subgroup of another.

the first is that it is non-empty.

well, is the intersection non-empty? is there an element that every subgroup of a group would have to have? if so, that element would be in the intersection of subgroups, right?

the second is closure under the operation.

well, is the intersection closed under the operation? take any two elements in the intersection, how would we know whether or not their "product" is in the intersection?

the last is the presence of inverses for each element. so take an element in the intersection. is its inverse in the intersection? how would you show that?

Q2) Let H = {(1), (1 2)} and K = {(1), (1 2 3), (1 3 2)}. Both H and K are subgroup of S3. Show that HUK is not a subgroup of S3. It follows that a union of subgroup is not necessarily a subgroup.
first thing's first. what is the set $H \cup K$?

all you need to show is that it violates one of the properties of being a sub-group. also note that subgroups are groups themselves. so if $H \cup K$ is not a group itself, that is also sufficient to prove it is not a subgroup. just go through and check the properties one by one. as soon as you find a violation, stop

5. 1) To show every element in the intersection has an inverse:
If a is in H ∩ K, then a is in H and a is in K. Since both of them are subgroups, a^-1 is in H and a^-1 is in K. Thus a^-1 is in H ∩ K.

2) What is (1 2 3) * (1 2)? Is it in H U K?...I thought union of that subgroup is a subgroup. Hence, since HUK is not in there...I don't know how to deal with??

6. Originally Posted by Vedicmaths
1) To show every element in the intersection has an inverse:
If a is in H ∩ K, then a is in H and a is in K. Since both of them are subgroups, a^-1 is in H and a^-1 is in K. Thus a^-1 is in H ∩ K.
that is correct. however, the are THREE things you need to show. you only showed one. look back at my (and your) posts. in addition to what you did, you need to show that the intersection is non-empty and that we have closure under the operation of G

2) What is (1 2 3) * (1 2)? Is it in H U K?...I thought union of that subgroup is a subgroup. Hence, since HUK is not in there...I don't know how to deal with??
(1 2 3)(1 2) = (1 3) ...what does that tell us?