Feasibility of groups as union of subgroups.

[seshikanth]

Homework Statement

I am trying to solve a question from Abstract Algebra by Hernstein.
Can anyone give me hint regarding the following:
Show that a group can not be written as union of 2 (proper) subgroups although it is possible to express it as union of 3 subgroups?

Thanks,

Homework Equations

The Attempt at a Solution

I know that the union of 2 subgroups is a group only when one is contained in another or viceversa.

 

[lanedance]

say you have G with proper subgroups A & B

consider some cases
case 1 - say A is contained in B
________but A & B are both proper

case 2 - say A & B are disjoint
________now consider the multiplication ab

case 3 - say A & B have elements in common other than e
________consider the set of elements they share

those cases should cover the range possibilities

 

[Dick]

If G is the union of two proper subgroups A and B then there is an element a of A that is not in B, and an element b of B that's not in A, right? I think that's the only case you need to consider. For the three subgroup case, just try and think of an example.

 

[lanedance]

trivial, but what if A is a subgroup of B?

 

[Dick]

trivial, but what if A is a subgroup of B?

If B is proper in G and A is a subgroup of B, then the union of A and B is B. Not G.

 

[lanedance]

yeah i suppose its really obvious, and the idea you give covers both 2 & 3

 

[lanedance]

as always, well played sir ;)

 

[Dick]

yeah i suppose its really obvious, and the idea you give covers both 2 & 3

That's the idea. If I were reading a students solution to this problem I'd be interested in the 'group theory' part. The case splitting would just annoy me.