Take the 2-minute tour ×
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.

Let $G$ be a group such that intersection of all its subgroups which are different from $(e)$ is a subgroup different from $(e)$. Prove that every element of $G$ has finite order.

Need hints to prove it or disprove it.

share|improve this question

2 Answers 2

up vote 5 down vote accepted

Suppose $a\in G$ is an element of non-finite order. Then the intersection of $A_d=\{a^{nd}, n\in Z\}$ is $e$, contradicting the premise.

share|improve this answer

Let $S$ be the smallest nontrivial subgroup of $G$, which exists by hypothesis, and let $a$ be any nonidentity element of $S$.

Show that $S=\langle a\rangle$, that $a$ has finite order, by contradiction, and that for any $x\in G$ we have $a\in\langle x\rangle$.

share|improve this answer
1  
"...the smallest non-trivial group..." +1 –  DonAntonio May 3 at 1:12

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.