User Tag List

Results 1 to 2 of 2

  1. #1
    Senior Member Bufo's Avatar
    Join Date
    Jul 2008
    MBTI
    onto
    Enneagram
    5w4
    Socionics
    INTp
    Posts
    122

    Default Is there a noncyclic group of order 39?

    The answer is no. But I'm supposed to solve this using basics of isomorphism (without any application of Sylow's theorem).

    Suppose such a group exists. Then of course it's non-Abelian.

    I know there are cyclic subgroups of order 3 or 13 (actually both, because possibly I'm allowed to use this particular implication of Sylow's theorem).
    One should not aim at being possible to understand, but at being impossible to misunderstand. - Quintilian

  2. #2

    Default

    If you're allowed to use that little tid-bit (seems redundant since LaGrange's theorem will give you that it will have one such cyclic supgroup if the whole group itself is not cyclic), then you are essentially done, because Z_3xZ_13 is isomorphic to Z_39. <(1,1)> generates the whole group.

    You still have to prove that since every group of order 39 has sub-groups of order 3 and of order 13, that it has to be isomorphic to Z_3xZ_13.

    It's been a while since I took group theory, but I think you can go about it by proving one or the other fo the supgroup of 3 or 13 is Normal in your original group. Then the factor group must be cyclic (again using LaGanges Theorem).

    That's just the sketch of a proof that popped into my head. But like I said, it has been a long time since my Math degree.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

Similar Threads

  1. Is there REALLY that much of a difference between MBTI and Keirsey?
    By Ezra in forum Myers-Briggs and Jungian Cognitive Functions
    Replies: 12
    Last Post: 08-29-2010, 01:23 AM
  2. Is there a function or a mix of functions that is responsible for this...
    By JustHer in forum Myers-Briggs and Jungian Cognitive Functions
    Replies: 2
    Last Post: 10-17-2009, 10:40 AM
  3. [INTP] Is there a common denominator to INTPs' line of interest?
    By Cypocalypse in forum The NT Rationale (ENTP, INTP, ENTJ, INTJ)
    Replies: 8
    Last Post: 04-09-2009, 11:50 AM

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
Single Sign On provided by vBSSO