Subgroup Q

[Sonal7]

Its a very minor question but it might underlie some misunderstanding of maths so I thought of asking for clarification.

is the identity element always a subgroup of any group? {e} ?
I thought it might be as there's closure and its a generator of all the elements in the group {e}?

The question is below I thought the answer to the last Q was {a,c} and also {a}, the answer says its only {a,c} (the answer says ONLY so I think they are trying to make a point of it). Thanks for any help, and sorry to be trivial ( Its not a pun).

 

[Sonal7]

I got it, the question asks for a proper subgroup, and {a} is a trivial subgroup! I got it when i wrote the word trivial, the pun solved it for me! Thanks anyway

 

[pka]

Here is a most useful theorem on subgroups: If \(\displaystyle \mathscr{G}\) is a group and \(\displaystyle H\subseteq\mathscr{G}\) then \(\displaystyle H\) is a subgroup of \(\displaystyle \mathscr{G}\) if and only if for each \(\displaystyle \{a,b\}\subset H\) then \(\displaystyle ab^{-1}\in H\).
Try to prove that. The proof is very instructive.

 

[Dr.Peterson]

What definition were you given for "proper subgroup"?

It sounds like you're saying that the trivial subgroup is not considered proper. See https://en.wikipedia.org/wiki/Subgroup,

A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).​

Presumably your book is written by "some author".

 

[Sonal7]

Our specification book says:

" If a non empty subset H of a group is itself a group under the binary operation of G, we call H a subgroup of G.
Every group has at least two groups (G,*) itself and ({e}*) . {e,*} is called a trivial subgroup, and any other subgroups called non trivial subgroup"

It also says B is a proper subgroup if it belongs to A but is not equal to A.

I guess my logic is wrong. I think its seems discretionary whether trivial subgroups are included in subgroups or not. hmmm. The book which follows the specification doesnt exclude trivial group from being proper, but the answer to one of integral maths website questions clearly does by saying 'ONLY' and then the non trivial subgroup. I think it maybe the person writing the answer booklet feel trivial groups are separate from proper subgroups . There are plenty of errors in general that I spotted on integral maths website that I sometimes ignore their answer and move on. Thanks, I don't think people should be this pendantic.

 

[Sonal7]

Here is a most useful theorem on subgroups: If \(\displaystyle \mathscr{G}\) is a group and \(\displaystyle H\subseteq\mathscr{G}\) then \(\displaystyle H\) is a subgroup of \(\displaystyle \mathscr{G}\) if and only if for each \(\displaystyle \{a,b\}\subset H\) then \(\displaystyle ab^{-1}\in H\).
Try to prove that. The proof is very instructive.

Its quite evident, I don't fully understand group theory. I need someone who does.

 

[Dr.Peterson]

I guess my logic is wrong. I think its seems discretionary whether trivial subgroups are included in subgroups or not. hmmm. The book which follows the specification doesnt exclude trivial group from being proper, but the answer to one of integral maths website questions clearly does by saying 'ONLY' and then the non trivial subgroup. I think it maybe the person writing the answer booklet feel trivial groups are separate from proper subgroups . There are plenty of errors in general that I spotted on integral maths website that I sometimes ignore their answer and move on. Thanks, I don't think people should be this pendantic.

This is one of the problems with internet learning: students who mix various sites, or mix online with their own textbook, too often stumble over these variations, just as they can stumble over dialects of English. Even in math, a word doesn't mean the same thing everywhere.

This is not primarily a matter of being pedantic (math has to define words carefully), but of people working in different fields, or just emphasizing different things, making different choices within their communities, based on what definition works best for them.

But the result is that you have to check the definitions used by any particular site, which most students will not think to do (and shouldn't have to). I recommend learning from one site only, or just from a textbook with a teacher who can answer questions based on its approach.

As to pka's comment: you don't need to fully understand group theory while you are first learning it. You just need one consistent source to learn from, so that you will know each theorem that a particular exercise is meant to give you practice with. His theorem is useful, but not necessarily one you should learn before other things. And it is not particularly relevant to the issue you asked about.