My Math Forum Abstract Algebra

 Abstract Algebra Abstract Algebra Math Forum

 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: 266
Thanks: 80

Math Focus: Algebraic Number Theory, Arithmetic Geometry
Quote:
 Originally Posted by shashank dwivedi 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?
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 Linear Mode

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

 Contact - Home - Forums - Cryptocurrency Forum - Top