Group Theory

Home Forums College Mathematics, GRE, TIFR Group Theory

This topic contains 5 replies, has 2 voices, and was last updated by  Sourayan Banerjee 2 months ago.

Viewing 5 posts - 1 through 5 (of 6 total)
  • Author
    Posts
  • #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 2 months, 1 week 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.

    Attachments:
    You must be logged in to view attached files.
    #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!

    Attachments:
    You must be logged in to view attached files.
    #30561

    SaSA :::::
    Participant

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

    Attachments:
    You must be logged in to view attached files.
Viewing 5 posts - 1 through 5 (of 6 total)

You must be logged in to reply to this topic.