Group Theory Basics: Where Can I Learn More?

[pdelong]

errata?

Good (introductory) references are:

M. A. Armstrong. Groups and symmetry. Springer Verlag 1988.

Would you happen to know where one could get an errata listing for this book? I'm currently in the middle of it and I'm stuck on a problem. I'm pretty sure the reason I'm stuck is because of a misprint, but I just want to be sure.

Thanks.

 

[Janitor]

I learned (well, sort of ) my group theory from a moldy-oldie library book written by _____ Hall. It was already old when I was 20, and I imagine it is long out of print.

 

[ahrkron]

On my computer, all of those are just roman letters drawn in a fancy way, not greek letters...

Same here! Mozilla on RedHat Linux. I'll check later on a Windows XP.

 

[ahrkron]

Same thing on XP (Mozilla also).

 

[meteor]

Groups are fantastic things, I'm recently starting to appreciate its importance, before I considered them rather arbitrary constructions
I'm wondering if the group operation is restricted to some binary operations, or instead, any binary operation is valid. Cause I know that for example addition, multiplication, matrix multiplication, composition of functions,... are binary operations that are permitted like group operations, but is this general?, I mean, can any binary operation serve like a group operation?

 

[matt grime]

No, by defintion the binary operation must satisfy certain rules, but by the same token any binary operation satisfying those rules is a group operation. "Serving like a group operation" is a nebulous phrase which could mean anything you chose it to mean.

A group is an axiomatic object, anything satisfying those axioms is a group, end of story. If you want a binary operation that isn't a group operation, there's multiplication on the real line - 0 has no inverse. Or addition on the set of multiples of 2 and 5 - that isn't closed, Addition on the strictly positive real numbers has no identity. For failure of asscoiativity I must be more creative: consider the group defined by this table:

* |a b
--------
a|a b
b|a b


That isn't associative by the failure of the latin square principle.

I forgot to emphasize that your question over looks the fact that there needs to be an underlying set the operation is defined on - matrix multiplication is not a group operation on the set of infinite matrices.

 

[meteor]

Matt,
Yeah, I know that a group has to satisfy certain rules (namely closure, associativity, identity element and existence of the inverse), but my question is: given a group of elements and a binary operation, and then those elements under the binary operation satisfy the rules, then this is considered a group, indepently of the binary operation? I ask this because I do not remember any group where the binary operation is division, or for example the modulus (a mod b), or the Legendre symbol, or many others, but would be nice if such groups could exist
Regards

 

[matt grime]

The binary operation is part of the group structure, it cannot be a group independently of the operation, hell it can't even be a group.

Given a set there are many ways of putting a group structure on it. There are two groups of order 6 that are not isomorphic.

I think you are misuing the word group in that clause 'given a group of elements'. Do you just mean set or group in its proper definition?

Division would not work for associativity reasons.

And those things you cite (legndgre symbol etc) cannot form group operations - one isn't even a binary operation.

 

[Janitor]

As Matt points out, division is not associative. Example: what is 20/10/2?

(20/10)/2 = 1,

20/(10/2) = 4.

 

[matt grime]

I misread what you wrote, they are all binary operations but cannot possibyl be group operations given, firstly you aren't offering a set on which they are to be defined which contradicts the definition that a group is a set with a binary operation, and they fail to define injective maps in that if you fix the first argument there are many (infinitely many) second arguments that will give the same output.

Remember, a group is a set WITH an operation satisfying... you need both.

 

[H-bar None]

There is the Dog School of Mathematics of dogpile fame. They have a nice tutorial on group theory.