Abstract Algebra Help

Discussion in 'Community Discussion' started by Shaun.P, Mar 12, 2009.

  1. Shaun.P macrumors 68000

    Shaun.P

    Joined:
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    Location:
    Omicron Persei 8
    #1
    Hi all,

    I was wondering if any maths people could help me with a question that I am stuck at that is really bothering me. [See picture below].

    I know the order of the set S15 is 15! (15 factorial).

    I also know how to prove the set H is a supgroup of S15.

    Can anyone tell me though, what the order of the subgroup is? Is it the same as the order of the set S15?

    Any help would be greatly appreciated.
     
  2. RHutch macrumors 6502

    Joined:
    May 21, 2003
    Location:
    Amsterdam, OH
    #2
    I have not done abstract algebra for several years, so there may be a better solution, but I think this is correct.

    In this case it is easy enough to directly compute the powers of alpha. Since there are only 6 permutations of three numbers (1, 14, 15), there are no more than 6 powers of alpha.

    All numbers except 1, 14, and 15 are fixed.
    Consider alpha squared: (1 14 15)(1 14 15).
    1 -> 14 -> 15
    14 -> 15 -> 1
    15 -> 1 -> 14

    So (1 14 15)(1 14 15) = (1 15 14)

    Then alpha cubed is (1 14 15)(1 15 14) and
    1 -> 15 -> 1
    15 -> 14 -> 15
    14 -> 1 -> 14

    So alpha cubed is the identity.

    So H = {alpha, alpha^2, and alpha^3}. The order of the subgroup H is 3.
     
  3. Shaun.P thread starter macrumors 68000

    Shaun.P

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    #3
    I never thanked you for your help.

    Thank you.

    I somehow managed to unsubscribe from this thread by mistake, and never noticed there was a reply.
     
  4. acearchie macrumors 68040

    acearchie

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    #4
    So is this a form of statistics or something completely different altogether?
     
  5. dukebound85 macrumors P6

    dukebound85

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    #5
    i covered it when i took "linear algebra and matrices" in college


    and yea, i forgot how to do that crap lol
     
  6. Shaun.P thread starter macrumors 68000

    Shaun.P

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    #6
    This is something completely different.

    The actual name of the course at my Uni is Algebraic Structures.

    This is the course plan:



    Mappings
    Composition
    Binary operations
    Composition as a binary operation
    Examples of groups
    Permutations
    Subgroups
    Equivalence relations
    Congruence. The division algorithm
    Integers modulo n
    Euclidean algorithm
    Definition and examples of rings
    Ideals
    Integral domains. Subrings
    Fields
    Isomorphism. Characteristic
    Ordered integral domains
    The integers
    Field of quotients
    Field of rational numbers
    Extensions of fields
    Definition and elementary properties of polynomials
    The division algorithm
    Factorisation of polynomials
    Unique factorisation domains
    Simple extensions
    Finite fields
    Partially ordered sets
    Lattices
    Boolean algebras
     
  7. jecapaga macrumors 601

    jecapaga

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  8. Frisco macrumors 68020

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    Utopia
    #8
    Wow Abstract Algebra is definitely a few steps above Calculus, which I was lucky to pass.
     
  9. emmawu macrumors 6502

    Joined:
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    Wauwatosa, WI
    #9
    The order of the subgroup H must be a divisor of 15! AND equals the minimal n such that alpha^n = alpha^n+1.

    Mambochicken (emmawu's so).
     
  10. RHutch macrumors 6502

    Joined:
    May 21, 2003
    Location:
    Amsterdam, OH
    #10
    I think you made a typo or some other error. Don't you mean that it is the minimal n for which alpha = alpha^(n + 1)? As you have written it, you would have consecutive powers of alpha equal to each other: n and n + 1.
     
  11. magamo macrumors 6502

    Joined:
    Apr 6, 2009
    #11
    Well, I don't want to be an ass, but since you corrected the error, I think the order is the minimal positive integer n >= 2 s.t. alpha = alpha^(n + 1) in this case. Otherwise, a smartass might say alpha = alpha^1 and hence the order has to be 0. An even more annoying ass could argue alpha = alpha^(-2) = alpha^(-5) =... because H happened to be the prime field of order 3.
     
  12. RHutch macrumors 6502

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    #12
    I'm confused; are you saying that I should not have commented on the powers?
     
  13. magamo macrumors 6502

    Joined:
    Apr 6, 2009
    #13
    No. Sorry, I expressed myself poorly... You're absolutely right about "alpha^n = alpha^n+1" is a typo or an error. I just wanted to make it complete because you kindly corrected part of emmawu's post.
     
  14. P-Worm macrumors 68020

    P-Worm

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    Salt Lake City, UT
    #14
    Guys, he specifically asked for Abstract algebra help. Why are we answering the question when it is intended for him? Where is Abstract, anyway?

    P-Worm
     
  15. Shaun.P thread starter macrumors 68000

    Shaun.P

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    #15
    Abstract algebra is the name of the branch of maths this is.
     
  16. jecapaga macrumors 601

    jecapaga

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    Southern California
    #16
    We will have to wait for his eventual arrival.
     
  17. emmawu macrumors 6502

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    Location:
    Wauwatosa, WI
    #17
    Hi,

    Mambochicken (a mathematics professor) admitted he made a typographical error. Maybe he should be taken to the parking lot and shot, huh? Or would that be too harsh?
     
  18. P-Worm macrumors 68020

    P-Worm

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    Jul 16, 2002
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    Salt Lake City, UT
    #18
    I know. I guess the joke didn't come out so well.

    P-Worm
     

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