December 22nd, 2017, 12:58 AM  #1 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0  Abstract Algebra
Prove that if G is a group such that G/Z(G) is cyclic, then G is abelian. (I am unable to connect logic)I don't know why? 
December 22nd, 2017, 02:25 AM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 203 Thanks: 60 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Say the coset of $Z(G)$ represented by $g \in G$ generates $G/Z(G)$. Then every coset of $Z(G)$ is represented by $g^k$ for some $k$. Take any two elements $a, b$ of $G$. $a$ must be contained in a coset of $Z(G)$, say it's in the one represented by $g^n$. This means $a = g^n x$ for some $x \in Z(G)$. Similarly, $b = g^m y$ for some $m$ and some $y \in Z(G)$. Now it's straightforward to show that $a$ and $b$ commute.


Tags 
abstract, algebra 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
abstract algebra q1  mohammad2232  Abstract Algebra  1  January 25th, 2015 06:37 AM 
Abstract Algebra Need Help  ustus  Abstract Algebra  4  October 14th, 2012 12:00 PM 
Abstract Algebra Help  MastersMath12  Abstract Algebra  1  September 24th, 2012 11:07 PM 
Abstract Algebra  forcesofodin  Abstract Algebra  10  April 5th, 2010 10:31 PM 
Abstract Algebra  micle  Abstract Algebra  0  December 31st, 1969 04:00 PM 