Not so: the union of two subgroups is almost never a subgroup (exercise: when exactly?).Originally Posted by ThePerfectHacker
You had better tell us what you want the expressions "union of two subgroups"Originally Posted by ThePerfectHacker
and "intersection of two subgroups" to mean. As it is they don't mean
anything as union and intersection are usually defined on sets. And a
groups is not a set, a set may be involved but the group is not the set,
and as rgep points out the naive union/intersection of the associated sets
of a pair of subgroups is not usually the set associated with another subgroup.
Maybe I did make a mistake. I just remember my book on Abstract Algebra asking to prove the intersection/uninon of groups, is a group something like that. Maybe, it is true for Normal Subgroups I do not remember exactly.
I do not think I made a mistake. By uninon/intersection I mean respectively. I just am visualizing that in my head, I think it is a subgroup.
But is not in general closed under the group operation, andOriginally Posted by ThePerfectHacker
so in general not a subgroup.
You might want to consider the group with the following group multiplication
table (which if I have done this right makes a group):
Then: , , , are all subgroups of ,
but is not a group because
is not closed under
Thank you CaptainBlack, my only mistake was that I assumed that addition was closed.
Okay now, change the problem, which type of subgroups will form a subgroup?
By the way, I like your group multiplication box, that probably took a long time in Latex code.