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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Which of the following statements is true? (V is the Klein-4 group)

    "=" represents isomorphic to.

    (A) V = R4
    (B) A3=R3
    (C) A4=R12
    (D) V =Z/4Z
    (E) (Z; +) = (Q; +).

    My answer was (A)..

    Please help
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  2. #2
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    Quote Originally Posted by Dreamer78692 View Post
    Which of the following statements is true? (V is the Klein-4 group)

    "=" represents isomorphic to.

    (A) V = R4
    (B) A3=R3
    (C) A4=R12
    (D) V =Z/4Z
    (E) (Z; +) = (Q; +).

    My answer was (A)..

    Please help

    What is R4, R3, R12, A3, A4...??

    Tonio
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  3. #3
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    Sorry about that...

    R - stands for the Rotation groups
    e.g. R3 = {e, r, r^2}

    A - Alternating Groups
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Dreamer78692 View Post
    Which of the following statements is true? (V is the Klein-4 group)

    "=" represents isomorphic to.

    (A) V = R4
    (B) A3=R3
    (C) A4=R12
    (D) V =Z/4Z
    (E) (Z; +) = (Q; +).

    My answer was (A)..

    Please help
    I presume by `rotation group' you mean D_{n}, the group of symmetries of a regular n-gon (Rn= D_n).

    You are correct. What were your thinkings in the other questions? (Order arguments and pointing out cyclic groups when you see them will work). Can you prove your result? (prove that D_4 is abelian but not cyclic).
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  5. #5
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    Quote Originally Posted by Swlabr View Post
    I presume by `rotation group' you mean D_{n}, the group of symmetries of a regular n-gon (Rn= D_n).

    You are correct. What were your thinkings in the other questions? (Order arguments and pointing out cyclic groups when you see them will work). Can you prove your result? (prove that D_4 is abelian but not cyclic).

    Very weird notation, but according to it, and if I didn't misunderstood, also (B) is correct since

    both A_3\,,\,R_3 are cyclic groups of order 3...

    Tonio
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    Very weird notation, but according to it, and if I didn't misunderstood, also (B) is correct since

    both A_3\,,\,R_3 are cyclic groups of order 3...

    Tonio
    Notation notation notation! However, I do not think you are correct in any notation. I write D_n for the set of symmetries of a regular n-gon, which is also often written as D_{2n}. It is always of even order, and so cannot be cyclic of order 3...

    It is true, however, that D_3 \cong S_3.
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  7. #7
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    Quote Originally Posted by Swlabr View Post
    Notation notation notation! However, I do not think you are correct in any notation. I write D_n for the set of symmetries of a regular n-gon, which is also often written as D_{2n}. It is always of even order, and so cannot be cyclic of order 3...

    It is true, however, that D_3 \cong S_3.


    The OP himself, though, wrote that R3 = \{e,r,r^2\}= a cyclic group of order 3. Perhaps for him "rotation group" is

    ONLY the actual rotations (spinnings) , no the whole dihedral group.

    Tonio
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  8. #8
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    The OP himself, though, wrote that R3 = \{e,r,r^2\}= a cyclic group of order 3. Perhaps for him "rotation group" is

    ONLY the actual rotations (spinnings) , no the whole dihedral group.

    Tonio
    Fair point. In which case, (A) is false.
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  9. #9
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    Quote Originally Posted by Swlabr View Post
    Fair point. In which case, (A) is false.

    Indeed, which I think was the case from the beginning, since then one is a cyclic group whereas the other isn't...who knows what the truth is here in Weirdnotationland.

    Tonio
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  10. #10
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    Sorry to go off topic for a moment, but I dream of a day when all mathematical notation and terminology is standardized internationally. Also, I wish they would stop referring to math as "maths" in the UK.
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  11. #11
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by MichaelMath View Post
    ...I wish they would stop referring to math as "maths" in the UK.
    That's okay, in the UK we wish they would stop referring to maths as `math' in the US.
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