Thread: Group Axioms

Hi!

Just want to clarify that the following is not a group because G2 fails

(Z,o), where x o y = 3x+y

G2 identity

e o x = 3x + e = x
that is 2x + e = 0
Take e = 0, then 0 is not an element of Z and so this is not a group.

Thanks.

Originally Posted by Arron

Hi!

Just want to clarify that the following is not a group because G2 fails

(Z,o), where x o y = 3x+y

G2 identity

e o x = 3x + e = x
that is 2x + e = 0
Take e = 0, then 0 is not an element of Z and so this is not a group.

Thanks.

You are correct that the problem is lack of inverse, however I think your argument is slightly flawed.

0 is not necessarily your identity. I mean, take Z under the operation a.b = a+b-1. Then 1 is your identity.

Here, you have that e=-2x for all , and so e is not fixed (and so no identity exists). Alternatively, you could note that so e=0, and so then you can use the argument you used above. But my point is that you need to prove that e=0, you can't just assume it is.

There are a few things worth mentioning... (I'll use "*" for the operation, FYI)
When you say:
"e * x = 3x + e = x",
you haven't correctly used the definition.
e * x := 3e + x

We want this to equal x, so e must be 0, which IS in Z! No contradiction (yet).

But if e = 0, then
x * e = 3x + e = 3x + 0 = 3x =/= x!

THAT is the problem!

Originally Posted by TheChaz

There are a few things worth mentioning... (I'll use "*" for the operation, FYI)
When you say:
"e * x = 3x + e = x",
you haven't correctly used the definition.
e * x := 3e + x

No-what he has said here is fine, as e*x=x because e is the identity (by assumption).

Originally Posted by Arron

Hi!

Just want to clarify that the following is not a group because G2 fails

(Z,o), where x o y = 3x+y

G2 identity

e o x = 3x + e = x
that is 2x + e = 0
Take e = 0, then 0 is not an element of Z and so this is not a group.

Thanks.

0 is an integer btw. Your argument could be [LaTeX ERROR: Convert failed] so [tex]e=-2x[/Math] but this is not fixed (dependant on x) which the identity must be therefore no identity element so not a group.

(I know what e is...)
e*x = x explains the second "=", not the first.
The definition of a*b is not symmetric!
His statement:
e * x = 3x + e = x

My revision:
e * x = 3e + x = x

Originally Posted by TheChaz

(I know what e is...)
e*x = x explains the second "=", not the first.
The definition of a*b is not symmetric!
His statement:
e * x = 3x + e = x

My revision:
e * x = 3e + x = x

Ah-touche-I didn't spot that error!

Thanks for your help everyone.