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Hello...
I need real help in these problems...
Please help!
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.
Thanks in advance!
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.
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.
you were right in what you said you needed to check if a group is a subgroup of another.Quote:
Originally Posted by Vedicmaths
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?
first thing's first. what is the setQuote:
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.
?
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
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??
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 GQuote:
Originally Posted by Vedicmaths
(1 2 3)(1 2) = (1 3) ...what does that tell us?Quote:
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??