Help with Group Theory notation

[grimjow7]

I have this problem: "Let G be a group and H≤G proper subgroup of G. Prove that <G−H> = G", I know that < > is a generator set, but I don't understand the subtraction notation "G-H". can someone help me to raise or solve the problem? thanks in advance.

 

[Steven G]

G-H is literally G-H. That is you have the set G and then from G you remove all the elements that are also in H. H does not have to even be a subset of G. Another way of thinking about it is G-H = [MATH]G\cap H'[/MATH]
Here is an example. Let G = {1, 2, 3} and H={1, 4, 5}. Then G-H = {2, 3}

 

[pka]

I have this problem: "Let G be a group and H≤G proper subgroup of G. Prove that <G−H> = G", I know that < > is a generator set, but I don't understand the subtraction notation "G-H". can someone help me to raise or solve the problem? thanks in advance.

In set-theory the notation \(G-H\) is called set difference more commonly \(G\setminus H\) is the set \(G\cap H^c\).
But in this case that cannot be it because the set \(G\setminus H\) does not contain the group identity.
You need to search the text/notes find the definition. Please post what you find.

 

[Steven G]

Yes, of course, if H < G then <G-H> can't be G using the definitions stated above.

 

[grimjow7]

G-H is literally G-H. That is you have the set G and then from G you remove all the elements that are also in H. H does not have to even be a subset of G. Another way of thinking about it is G-H = [MATH]G\cap H'[/MATH]
Here is an example. Let G = {1, 2, 3} and H={1, 4, 5}. Then G-H = {2, 3}

Thank you so much, I'm starting a group theory course and I was not sure whether to interpret it textually as the difference of sets or as some set formed by differences of elements, because I did not know if there was something like group algebra.

 

[pka]

I have this problem: "Let G be a group and H≤G proper subgroup of G. Prove that <G−H> = G", I know that < > is a generator set, but I don't understand the subtraction notation "G-H". can someone help me to raise or solve the problem? thanks in advance.

Thank you so much, I'm starting a group theory course and I was not sure whether to interpret it textually as the difference of sets or as some set formed by differences of elements, because I did not know if there was something like group algebra.

Are you absolutely sure that the notation is not \(\left<G/H\right>~\).
I have searched quite a few standard modern algebra textbooks and found your notation in none of them.
Of course, all that means is that I do not know your texts. So please give us more information on the source of this question.

 

[grimjow7]

In set-theory the notation \(G-H\) is called set difference more commonly \(G\setminus H\) is the set \(G\cap H^c\).
But in this case that cannot be it because the set \(G\setminus H\) does not contain the group identity.
You need to search the text/notes find the definition. Please post what you find.

Indeed, G-H does not contain the identity group, but <G-H> can generate the identity. Taking x in G-H, x ^ 1 is not in H because if it were, x would be in H, so taking the product of xx ^ 1 = e, it is possible to generate it with the conjugate G-H, that is <G-H>

Are you absolutely sure that the notation is not \(\left<G/H\right>~\).
I have searched quite a few standard modern algebra textbooks and found your notation in none of them.
Of course, all that means is that I do not know your texts. So please give us more information on the source of this question.

Precisely I did not find the notation "-" so I was in doubt, it is an exercise in my group theory course, possibly the person who wrote the exercises did not take into consideration the usual group notation, but everything seems to indicate that it is indeed \(\left<G/H\right>~\).

 

[Dr.Peterson]

As others have said, you need to look in your own textbook to see how they define their notation.

But I searched for a similar statement, and found this, which solves your problem (4a), and does mean set difference (G\H, not G/H!). The key is the notation <A>, which is not the generator set as you said, but the generated set.

https://www.tau.ac.il/~ginzburg/alg_b1_b_10.pdf