Generators and relation question

nhk

Apr 2010
43
0
What is the minimum number of generators needed for Z2+Z2+Z2? Find a set of generators and relations for this group.
 
Oct 2009
4,261
1,836
What is the minimum number of generators needed for Z2+Z2+Z2? Find a set of generators and relations for this group.

Three (3): \(\displaystyle \mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2=\left\{a,b,c\;;\;a^2=b^2=c^2=1\,,\,\,[a,b]=[a,c]=[b,c]=1\right\}\) (change the powers to coefficients if you prefer an additive notation).

With the "obvious" definitions , the above group is a three-dimensional vector space over the field \(\displaystyle \mathbb{F}_2:=\mathbb{Z}_2:=\mathbb{Z}/2\mathbb{Z}\) .

Tonio
 
  • Like
Reactions: nhk

nhk

Apr 2010
43
0
How would I prove that the minimum number of generators is 3? ( I think this is probable a stupid question, but my book does not explain anything about this generator stuff!!). i also do not understand to [a,b]=[a,c]=[b,c]=1 part. Thanks for your help.
 
May 2009
1,176
412
How would I prove that the minimum number of generators is 3? ( I think this is probable a stupid question, but my book does not explain anything about this generator stuff!!). i also do not understand to [a,b]=[a,c]=[b,c]=1 part. Thanks for your help.
By definition, \(\displaystyle [g, h]:=g^{-1}h^{-1}gh\). This is called the commutator. Note that two elements \(\displaystyle g\) and \(\displaystyle h\) commute if and only if \(\displaystyle [g, h] = 1\).

So, what they are saying is that the three generators all commute. In general, if \(\displaystyle G_1 = <X_1 ; R_1>\) and \(\displaystyle G_2 = <X_2 ; R_2>\) then \(\displaystyle G_1 \times G_2 = <X_1, X_2 ; R_1, R_2, [X_1, X_2]>\) because the direct product (although in your question it is the diect sum, but that is fine) means that the elements of each individual group commute with all the others. This holds if and only if the generators commute from the respective groups.

Now, as for the number of generators, note that every element has order two and your group is of order 8 and abelian. This means that it cannot be 2-generated. However, I will leave it to you to show that these facts also means that it cannot be 2-generated (a quick proof by contradiction will suffice).
 
  • Like
Reactions: nhk
Oct 2009
4,261
1,836
By definition, \(\displaystyle [g, h]:=g^{-1}h^{-1}gh\). This is called the commutator. Note that two elements \(\displaystyle g\) and \(\displaystyle h\) commute if and only if \(\displaystyle [g, h] = 1\).

So, what they are saying is that the three generators all commute. In general, if \(\displaystyle G_1 = <X_1 ; R_1>\) and \(\displaystyle G_2 = <X_2 ; R_2>\) then \(\displaystyle G_1 \times G_2 = <X_1, X_2 ; R_1, R_2, [X_1, X_2]>\) because the direct product (although in your question it is the diect sum, but that is fine) means that the elements of each individual group commute with all the others. This holds if and only if the generators commute from the respective groups.

Now, as for the number of generators, note that every element has order two and your group is of order 8 and abelian. This means that it cannot be 2-generated. However, I will leave it to you to show that these facts also means that it cannot be 2-generated (a quick proof by contradiction will suffice).


Hmmm...if the OP doesn't know the commutator notation I can't understand how she/he is trying to cope with presentations/generators/relations questions, which is usually studied after a basic course in group theory.

Anyway, and i shoty: as the hint in my first post "hinted", the fact that the given group is in fact an elementary finite abelian group makes it clear that its minimal number of generators = its dimension as avector space over some finite field...
Another easy proof could also use the Fratinni subgroup of G since the given group is a 2-group.

Tonio
 
  • Like
Reactions: nhk
May 2009
1,176
412
Anyway, and i shoty: as the hint in my first post "hinted", the fact that the given group is in fact an elementary finite abelian group makes it clear that its minimal number of generators = its dimension as avector space over some finite field...
I decided not to mention that proof as I was unsure if they would have covered this. Needless to say, my grasp of when concepts are encountered is somewhat lacking...

That said, I have been reading a book recently called "Groups, Graphs and Trees". It's excellent, because it covers a lot of useful things, but in a really easy to read way because it's an undergrad book, it was taken from an undergrad course given by someone! It covers things like ends and introduces the Coxeter groups, and I was just flicking through it and I saw a wreath product! My point is, people cover weird things in the undergrads...
 
Oct 2009
4,261
1,836
I decided not to mention that proof as I was unsure if they would have covered this. Needless to say, my grasp of when concepts are encountered is somewhat lacking...

That said, I have been reading a book recently called "Groups, Graphs and Trees". It's excellent, because it covers a lot of useful things, but in a really easy to read way because it's an undergrad book, it was taken from an undergrad course given by someone! It covers things like ends and introduces the Coxeter groups, and I was just flicking through it and I saw a wreath product! My point is, people cover weird things in the undergrads...

I have the book "Groups, Graphs and Trees", by John Meier. I don't know why you think it is an undergraduate one. It certainly covers some basic stuff, just like free groups and actions, but only shortly and it gets into deep waters right away.
This book could perhaps be covered in a seminar course for advanced undergraduates-graduates, but I doubt highly it could be an open course for unders in most schools I know of.

Tonio
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
I decided not to mention that proof as I was unsure if they would have covered this. Needless to say, my grasp of when concepts are encountered is somewhat lacking...

That said, I have been reading a book recently called "Groups, Graphs and Trees". It's excellent, because it covers a lot of useful things, but in a really easy to read way because it's an undergrad book, it was taken from an undergrad course given by someone! It covers things like ends and introduces the Coxeter groups, and I was just flicking through it and I saw a wreath product! My point is, people cover weird things in the undergrads...
I have the book "Groups, Graphs and Trees", by John Meier. I don't know why you think it is an undergraduate one. It certainly covers some basic stuff, just like free groups and actions, but only shortly and it gets into deep waters right away.
This book could perhaps be covered in a seminar course for advanced undergraduates-graduates, but I doubt highly it could be an open course for unders in most schools I know of.

Tonio
Just to add in, I saw the commutator quite a bit as an undergrad (which I guess is right now) and we even covered minimally basic examples regarding generators and relations.
 
May 2009
1,176
412
I have the book "Groups, Graphs and Trees", by John Meier. I don't know why you think it is an undergraduate one. It certainly covers some basic stuff, just like free groups and actions, but only shortly and it gets into deep waters right away.
This book could perhaps be covered in a seminar course for advanced undergraduates-graduates, but I doubt highly it could be an open course for unders in most schools I know of.

Tonio
The second last paragraph of the preface,

"Groups, Graphs and Trees was developed from notes used in two undergraduate course offerings at Lafayette College, and can certainly serve as a primary text for an advanced undergraduate course."
 
Oct 2009
4,261
1,836
The second last paragraph of the preface,

"Groups, Graphs and Trees was developed from notes used in two undergraduate course offerings at Lafayette College, and can certainly serve as a primary text for an advanced undergraduate course."

"From two under. courses" seems to be the key, although I still believe this is just one example more of overstatement of his own work by an author.
Here's the list of under. courses at Lafayette College:
Lafayette:Math:Courses

Making a quick check reveals they don't even have any course in group theory, and the theory covered by Abstract Algebra I-II seems to be below the average of what's covered in other places.

So perhaps some teacher did more group theory in some course(s) and that's what the author meant, though it is highly doubtable that most of the material could be gathered from under. courses.

Tonio