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Math Help - Does the identity element form a group on it's own?

  1. #1
    Ant
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    Does the identity element form a group on it's own?

    Hi,

    Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?

    Am I correct in thinking that it would:

    1) Identity element exists by assumption
    2) All g in G have an inverse (the identity is its own inverse)
    3) composition under the operation of the identity is associative

    Thanks!
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    Super Member TheChaz's Avatar
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    Quote Originally Posted by Ant View Post
    Hi,

    Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?

    Am I correct in thinking that it would:

    1) Identity element exists by assumption
    2) All g in G have an inverse (the identity is its own inverse)
    3) composition under the operation of the identity is associative

    Thanks!
    Yes. It's called the trivial group.
    Also note that any subgroup of a (different/larger) group must contain "e".
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Ant View Post
    Hi,

    Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?

    Am I correct in thinking that it would:

    1) Identity element exists by assumption
    2) All g in G have an inverse (the identity is its own inverse)
    3) composition under the operation of the identity is associative

    Thanks!
    You forgot closure.

    A subgroup consisting only of the identity is, along with the "subgroup" created by all elements of the group itself, is called a "trivial subgroup." Yes, it is a subgroup.

    A "proper subgroup" is a subgroup that is not just the identity, nor the entire group.

    -Dan
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    Quote Originally Posted by topsquark View Post
    You forgot closure.

    A subgroup consisting only of the identity is, along with the "subgroup" created by all elements of the group itself, is called a "trivial subgroup." Yes, it is a subgroup.

    A "proper subgroup" is a subgroup that is not just the identity, nor the entire group.

    -Dan

    Is this common terminology outside of pure mathematics? I have never once seen an author exclude the trivial sub-object from the class of proper sub-objects, and I have likewise never seen the object itself referred to as a trivial sub-object.
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  5. #5
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    it is quite common yes. for example, excluding the trivial subgroup as a proper subgroup allows the characterization of a simple group as a group with no proper normal subgroups, rather than a group with no nontrivial proper normal subgroups. however, this can vary from author to author.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Ghost View Post
    Is this common terminology outside of pure mathematics? I have never once seen an author exclude the trivial sub-object from the class of proper sub-objects, and I have likewise never seen the object itself referred to as a trivial sub-object.
    In Physics it depends. "Standard" Physics, that is to say the usual mixture of theory and experimentation, does not typically make a distinction. Mathematical Physics on the other hand is much more thorough, even if not as much as Mathematics. My Math Physics books make the distinction, my graduate Quantum Physics book (the "standard text" in the field) does not.

    -Dan
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