Group Theory

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  • #29957
    SaSA :::::
    Participant

    Let G be a finite group in which (ab)^p=a^pb^p for all a,b in G, where p is a prime dividing order of G. Then prove that G has a non-trivial center.
    (This is part c of problem 15 in section 2.12, of Herstein’s topics in algebra.)

    • This topic was modified 10 months, 2 weeks ago by SaSA :::::.
    #30224
    Sourayan Banerjee
    Participant

    Please have a look at Direct Product of groups. Solution is attached but I have left some obvious exercises, give those a try.

    #30227
    Sourayan Banerjee
    Participant

    Sorry for the typos out there I am attaching a pdf. That’ll help.

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    #30554
    SaSA :::::
    Participant

    I hadn’t read about direct products up to this problem so it didn’t come to me to use them, and I completely missed that solution of part two could be used. Thanks a lot for your solution, using it I was able to find an alternative which avoids direct products, essentially the idea is to show that the center of P is in the center of G, which is also what your solution does, thanks again!

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    #30561
    SaSA :::::
    Participant

    There are some unnecessary arguments in my previous attachment, please refer this, it’s better.

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