Tagged: Direct Product of Groups
- July 12, 2019 at 9:41 pm #29957
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.)
July 14, 2019 at 8:53 pm #30224Sourayan BanerjeeParticipant
- This topic was modified 10 months, 2 weeks ago by SaSA :::::.
Please have a look at Direct Product of groups. Solution is attached but I have left some obvious exercises, give those a try.
July 14, 2019 at 9:04 pm #30227Sourayan BanerjeeParticipant
- This reply was modified 10 months, 2 weeks ago by Sourayan Banerjee.
Sorry for the typos out there I am attaching a pdf. That’ll help.July 16, 2019 at 5:14 pm #30554
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!July 16, 2019 at 7:18 pm #30561
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