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Math Help - counting inverses of finite group

  1. #1
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    counting inverses of finite group

    Hi

    I am using Charles Pinter's book on abstract algebra . I have to prove that
    for any finite group G, if we define the set

    S=\{x\in G : x\neq x^{-1} \}

    i.e. set of elements in G which are not equal to its own inverse. I have to prove that
    set S has even no of elements. Now I can see that if some x is in S then its inverse
    has to be there since

    (x^{-1})^{-1}\neq x^{-1}

    so members of S appear in pairs. So I can see that , its cardinality has to be even.
    But how can I prove this ? any hints ?

    thanks
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  2. #2
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    Re: counting inverses of finite group

    what's left to prove?
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    Re: counting inverses of finite group

    Hi Deveno, I just wanted to know how to put this in mathematical language ? more formal language when you write the proof...
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    Re: counting inverses of finite group

    Quote Originally Posted by issacnewton View Post
    I just wanted to know how to put this in mathematical language ? more formal language when you write the proof...
    Suppose that \{x,y\}\subseteq\mathcal{S}~\&~x\ne y,~x\ne y^{-1} then is it true that \left\{ {x,x^{ - 1} } \right\} \cap \left\{ {y,y^{ - 1} } \right\} = \emptyset~?

    Is it true that \mathcal{S}=\bigcup\limits_{x \in S} {\left\{ {x,x^{ - 1} } \right\}}~?

    What does that tell about the cardinality of  \mathcal{S} ~?
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    Re: counting inverses of finite group

    Quote Originally Posted by issacnewton View Post
    Hi Deveno, I just wanted to know how to put this in mathematical language ? more formal language when you write the proof...
    since x^{-1} \in S whenever x \in S and for all x \in S,\ x \neq x^{-1}, the elements of S occur in distinct pairs.

    so if S has k such pairs, |S| = 2k, which is even.

    unless you are required to put this in the form of a well-formed statement in some first-order formal language, that is all you need say.
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    Re: counting inverses of finite group

    Thanks Deveno.......thats better...

    Plato, its true that

    \left\{x,x^{-1}\right\}\cap \left\{y,y^{-1}\right \}=\emptyset

    so we can say that cardinality of S is some power of 2 since each subset
    \{x,x^{-1}\} of S has two distinct elements.... hence
    the proof........right ?
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    Re: counting inverses of finite group

    Quote Originally Posted by issacnewton View Post
    Hi Deveno, I just wanted to know how to put this in mathematical language ? more formal language when you write the proof...
    As others have suggested, there's really nothing more to do once you notice that S is composed of pairs of inverses. I can understand the desire to express the point in a more computational manner, but that will only make the proof harder to write---and more importantly, harder to read.

    By the way, an interesting result from considering S is that we can prove Cauchy's theorem for the case p=2. For if G is a group of even order, then S and the identity element make up an odd number of elements, which means there must be at least one non-identity element in G which is its own inverse, i.e. has order 2.
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    Re: counting inverses of finite group

    Quote Originally Posted by issacnewton View Post
    Thanks Deveno.......thats better...

    Plato, its true that

    \left\{x,x^{-1}\right\}\cap \left\{y,y^{-1}\right \}=\emptyset

    so we can say that cardinality of S is some power of 2 since each subset
    \{x,x^{-1}\} of S has two distinct elements.... hence
    the proof........right ?
    Not a power of 2... a multiple of 2.
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  9. #9
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    Re: counting inverses of finite group

    Thanks everybody.

    and hatsoff, thanks for the additional input about Cauchy's theorem.....

    I just wanted to know more rigorous mathematical way of presenting what I thought. I had taken a class on introductory real analysis few years ago. And the teacher was very strict about presenting logical arguments. Since it was an analysis class , our arguments had to be logically watertight. Before taking the
    analysis class, I used think that some things are obvious. During this class, in the beginning I lost many points on assignments because my arguments
    were not as watertight. I think a class in real analysis really forces you to think logically. May be other branches of mathematics dont force such
    mental habits. I am just trying to be loyal to the rigor as demanded in real analysis.

    thanks
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