Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $G$ be a finite group with more than one element. Show that $G$ has an element of prime order

share|cite|improve this question
    
Does Z4 (the mod 4 group) have an element of prime order? The identity has order 1, and the other elements have order 4. – barrycarter Jan 25 '13 at 14:01
    
@barrycarter No, $2$ has order $2$. – SpamIAm Jun 27 '15 at 14:32
    
I can't believe I made such a simple error. – barrycarter Jul 4 '15 at 3:08

Hint: Let $a \neq 1$ be an element of $G$ of order $k$ and $d \in \mathbb{N}$ be a number such that $d \mid k$. What is the order of $a^d$?

share|cite|improve this answer
    
+1 You really show the OP how to find that element in $G$. ;-) – S. Snape Jan 25 '13 at 8:59

Let $|G|=n$ and $p\mid n$ such that $p$ be a prime , so it is enough to apply Cauchy's Theorem and find that desired element.

share|cite|improve this answer
    
Nice observation! +1 – amWhy Feb 2 '13 at 0:16
    
@amWhy: Thanks for your kindness. – S. Snape Feb 2 '13 at 6:25

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.