May 30th, 2017, 12:46 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  True or False
In a Group $G$ if $\displaystyle a \in G$ , $a^7=e$ and $a^9=e$ then $a=e$ ? I guess it is true. Someone help! Thank you 
May 30th, 2017, 01:28 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1383 
$a = e^5a = (a^7)^5a = a^{36} = (a^9)^4 = e$

May 30th, 2017, 04:55 AM  #3 
Member Joined: Jan 2016 From: Athens, OH Posts: 46 Thanks: 26 
In a group, if $a^n=e$, then the order of a divides n  if d is the order write $n=dq+r\text{ with }0\leq r<d$. So $e=a^{ndq}=a^r$. Thus r=0. So if $a^m=a^n=e$, the order of a divides the gcd of m and n. In particular, if m is prime to n, the order of a is 1; i.e. a=e. 

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