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Question about a Theorem in Gallian's Contemporary Abstract Algebra

  1. Oct 11, 2008 #1
    Question about a Theorem in Gallian's "Contemporary Abstract Algebra"

    I'm using this book as a reference for my Algebra course, and there's a lemma in the book that is really confusing me.

    It is on Page 102 of the Sixth Edition, for those who have the book.

    The lemma states:

    If [tex]\epsilon=\beta_1\beta_2\cdots\beta_r[/tex] where the [tex]\beta 's[/tex] are 2-cycles, then r is even.

    The author states that it is a special case of the theorem which says: if a permutation A can be expressed as a product of an even(odd) number of 2-cycles, then every decomp. of A into a product of 2-cyles must have an even(odd) number of 2-cycles.


    But doesn't the lemma state that every cycle can be written as a product of an even number of two cycles? I'm confused, and I'm not following the proof of the lemma.

    Any help would be appreciated.
     
  2. jcsd
  3. Oct 11, 2008 #2
    Re: Question about a Theorem in Gallian's "Contemporary Abstract Algebra"

    I apologize. Turns out that [tex]\epsilon[/tex] is the identity permutation.
     
  4. Oct 12, 2008 #3

    JasonRox

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    Re: Question about a Theorem in Gallian's "Contemporary Abstract Algebra"

    Haha, I saw your post and had the book out (see my thread) and I was thinking the same thing at first glance, but then it's like... wait a second, e is the identity. It's not written like the typical e like the rest of the book, so I can understand the confusion.
     
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