A Question on Semantics Regarding Group Theory

[Mindstein]

Homework Statement

Is the set of a single element {e} with the multiplication law ee = e a group?

Homework Equations

none.

The Attempt at a Solution

Yes, it is a group. But that is not my question. My question is how do you ask the question? If I were face to face with you and wanted to ask you the question, would I say, "Is the set of a single element e with the multiplication law e multiplied by ANOTHER e (another element in the group) equivalent to the identity element?"

Also, if I am wrong about it being a group...who cares. If I get the semantics down first, I will better understand what the problem is asking.

Thanks everyone!

 

[oli4]

Hi Mindstein.
I am not sure I get your question.
{e} is a group (for say, multiplication) IF e is the neutral element (or identity).
for any group whose neutral element is e, {e} is (with the group itself) the trivial subgroup.
What do you mean 'another e' ?
If you are thinking that you can take any element of the original group and take put it in a sngleton and wonder if this singleton is also a group, than the answer is no. it's only valid for e.
for instance, take (N, +), 0 is its 'e', so {0} is a group, but {1} is not since 1+1 does not belong to {1}
sorry if I didn't get your question

cheers...

 

[tiny-tim]

Hi Mindstein!

Is the set of a single element {e} with the multiplication law ee = e a group?

Yes.

If you're worried that you can't pick two e's in S, it doesn't matter …

the law about multiplication is defined on S x S, not on S itself (where S is the set),

and (e,e) is an element of S x S ​

 

[Mindstein]

Thanks tiny-tim and oli4, you all sure do know how to get a brother past his problems!