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!

 

[paranormal]

I appreciate your attention to detail and clarity in asking the question. To answer your first question, you could ask, "Is the set containing only the identity element, with the operation defined as e multiplied by another element in the set, a group?" This phrasing makes it clear that the operation is defined within the set itself.

In terms of whether this set is a group or not, it depends on the definition of a group you are using. Some definitions require a group to have at least two elements, while others allow for a single element group. However, in this case, since the operation is defined as e multiplied by another element in the set, it is not a valid group according to the standard definition. This is because for a group, the operation must be closed, meaning that the result of the operation must always be within the set. In this case, the result of e multiplied by another element (since e is the only element) would not be within the set, as it would be a different element.

I hope this helps clarify the semantics and the mathematical concept in question. It is always important to understand the context and definitions when approaching a problem in science.