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Subgroup of Dihedral Group

  1. Jan 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Taking the Dih(12) = {α,β :α6 = 1, β2 = 1, βα = α-1β}
    and a function Nr = {gr: g element of Dih(12)}



    2. Relevant equations
    Taking the above I have to find the elements of N3. And then prove that N3 is not a subgroup of Dih(12).


    3. The attempt at a solution
    For N3 I have determined the following are elements. 1, α3, β, βα3.
    Now to attempt to prove whether these elements form a subgroup of Dih(12) I created the multiplication table. The result was the table was closed and so I would assume that the aforementioned subset is a subgroup.
    I assume I have done one of two things wrong; either
    a) My understanding of generating n3 is misguided, though if I take an element of Dih(12), say αβ, and then using the function defined I get (αβ)3 = α3ββ = α3.

    or b) my multiplication table is wrong.

    Any help on what my mistake is is appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 12, 2012 #2

    micromass

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    I don't get how you did this calculation. Could you show more steps??
     
  4. Jan 12, 2012 #3
    I think you did (αβ)^3 = α^3β^3 but actually (αβ)^3=αβαβαβ

    so say (αβ)^3= g then αβαβαβ=g so βαβαβ=α^-1g

    thus βαβαβ=βαβg so g=αβ
     
  5. Jan 13, 2012 #4
    in reply to micromass...
    (σβ)3 I did: αβαβαβ = αααβββ = α3β3 = α3β (as ββ = 1)
    Now, if I follow what conquest has said I should have (and recalling βα = α-1β)
    (σβ)3 = αβαβαβ = αα-1α-1β = 1α-1β = βα.
    and for some of the other elements I get
    2β)3 = α2βα2βα2β = α2α-2ββα2β = α2β
    3β)3 = α3βα3βα3β = α3α-3ββα3β = α3β

    therefore If I apply the same understanding as above to every element of Dih(12) I get the following results for N3
    ....
    So the elements of N3 are
    1, α3, β, βα, βα2, βα3, βα4, βα5

    and if I do out the multiplication table all elements of N3 are contained in the table, along with the following elements that are not elements of the set N3
    α, α2, α4, α5
    hence N3 is not a subgroup of Dih(12).

    Am I going on the right tracks with this?
     
  6. Jan 13, 2012 #5
    I should also add that for
    (α)3 = ααα = α3
    2)3 = α2α2α2 = α6 = 1
    3)3 = α3α3α3 = α9 = α6α3 = α3
    4)3 = α4α4α4 = α12 = α6α6 = 1
    5)3 = α5α5α5 = α15 = α12α3 = α3
     
  7. Jan 13, 2012 #6
    exactly for instance:

    β³=β
    (αβ)³=αβ

    and βαβ=α-1= α5

    which is not in N3
     
  8. Jan 13, 2012 #7
    Excellent, many thanks!
     
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