Home › Forums › College Mathematics, GRE, TIFR › Group Theory

Tagged: Direct Product of Groups

- This topic has 5 replies, 2 voices, and was last updated 10 months, 2 weeks ago by Sourayan Banerjee.

- AuthorPosts
- July 12, 2019 at 9:41 pm #29957SaSA :::::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 :::::.

July 14, 2019 at 8:53 pm #30224Sourayan BanerjeeParticipantPlease have a look at Direct Product of groups. Solution is attached but I have left some obvious exercises, give those a try.

- This reply was modified 10 months, 2 weeks ago by Sourayan Banerjee.

July 14, 2019 at 9:04 pm #30227Sourayan BanerjeeParticipantSorry for the typos out there I am attaching a pdf. That’ll help.

###### Attachments:

You must be logged in to view attached files.July 16, 2019 at 5:14 pm #30554SaSA :::::ParticipantI 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.July 16, 2019 at 7:18 pm #30561SaSA :::::ParticipantThere are some unnecessary arguments in my previous attachment, please refer this, it’s better.

###### Attachments:

You must be logged in to view attached files. - AuthorPosts

- You must be logged in to reply to this topic.