My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum

LinkBack Thread Tools Display Modes
June 23rd, 2016, 05:13 AM   #1
Joined: Oct 2013

Posts: 36
Thanks: 1

H < Z(G) and G/H nilpotent, implies G is nilpotent.

So I am doing this question from Rotman (pg 117, q5.38 ): if H is a subgroup of the center of G and G/H is nilpotent then G is nilpotent.

I have looked at other proofs on other websites but they do not match what I have down so I just wanted to check if what I did was right too.

I said let $|G| = p_1^{e_1}...p_n^{e_n}|H|, p_i$ are primes.
Then $G/H \cong P_1 \times ... \times P_n$ where $P_i$ is a Sylow $p_i$-subgroup of order $p_i^{e_i}$ since $G/H$ is nilpotent.
Now since $H$ is abelian it is nilpotent too but also we can get an isomorphism from $Q_1 \times ... \times Q_n \times H$ to $G$ where $Q_i$ is the preimage of $P_i$ in $G$ by the natural map.
Then since $Q_1 \times ... \times Q_n$ and $H$ are both nilpotent so is $G$ as their direct product.

Does this work?

Edit: damn realised this only works for finite G- in any case is this still valid if G is finite?

Last edited by fromage; June 23rd, 2016 at 05:22 AM.
fromage is offline  
June 25th, 2016, 03:58 AM   #2
Senior Member
Joined: Sep 2008

Posts: 150
Thanks: 5

The line, were you state the isomorphism of G with a Group that has H as a factor is wrong: A counterexample would be G cyclic of order 4 and H the unique subgroup of order 2.

I would use the characterization of nilpotent with the Central series to prove your claim.
Peter is offline  

  My Math Forum > College Math Forum > Abstract Algebra

&lt, <, g or h, implies, nilpotent

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Nilpotent elements in rings PeterMarshall Abstract Algebra 2 January 20th, 2012 01:15 AM
nilpotent tinynerdi Abstract Algebra 2 April 7th, 2010 11:46 PM
Can you help me?... nilpotent group... vananh Abstract Algebra 0 April 9th, 2009 01:31 AM
Nilpotent matrices angelz429 Linear Algebra 2 May 7th, 2008 07:39 AM
locally-nilpotent group sastra81 Abstract Algebra 2 February 24th, 2007 02:32 AM

Copyright © 2017 My Math Forum. All rights reserved.