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Math Help - Quick question about identity element

  1. #1
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    Quick question about identity element

    Say I have a group G and an unknown binary operation *, and a*a=e, \forall a \in G. ( e being the identity element)

    Is it true that the only value a can be is the identity element, e?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Quick question about identity element

    Quote Originally Posted by tangibleLime View Post
    Say I have a group G and an unknown binary operation *, and a*a=e, \forall a \in G. ( e being the identity element). Is it true that the only value a can be is the identity element, e?
    Not true. Choose for example (\mathbb{Z}_2,+) , we have 0+0=1+1=0 .
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    Re: Quick question about identity element

    Ah, didn't consider that. Thanks for the response!
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    Re: Quick question about identity element

    what you can say, if a*a = e,and a is NOT the identity, is that a is of order 2, and that a is its own inverse (showing that elements of order 2 and of order 1(the identity is the only element of order 1) share this property).

    a good example of an element of order 2 is the following group, which is a subset of the actions you can perform on a light switch:

    {flip the switch, do nothing} and the group operation is: first do one thing, then the other.

    "flip the switch" is an element of order 2: flip the switch, then flip the switch = do nothing.

    (to be honest, this is in all actuality just Z2 in disguise, and is one of the basic ways in which logical circuits can "do math").
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    MHF Contributor Swlabr's Avatar
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    Re: Quick question about identity element

    What is true, however, is that if a\ast a=a then a is the identity. This is true because,

    a=a^{-1}\ast a\ast a=a\ast a^{-1}=e
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