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Math Help - Group Axioms

  1. #1
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    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.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Arron View Post
    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 x \in \mathbb{Z}, and so e is not fixed (and so no identity exists). Alternatively, you could note that e\circ y = e+y=y 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.
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  3. #3
    Super Member TheChaz's Avatar
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    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!
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by TheChaz View Post
    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).
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  5. #5
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    Quote Originally Posted by Arron View Post
    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.
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  6. #6
    Super Member TheChaz's Avatar
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    (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
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  7. #7
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by TheChaz View Post
    (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!
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  8. #8
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    Thanks for your help everyone.
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