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.

a)find a nontrivial group $G$ such that $G$ is isomorphic to $G \times G $

what i'm sure is that $G$ must be infinite ! but have now idea how to get or construct such group i chose many $G$'s but all of the homomorphism was not injective

b) an infinite group in which every element has finite order but for each positive integer n there is an element of order n

the group $G = (Z_1 \times Z_2 \times Z_3 \times Z_4 \times ...) $ satisfies the conditions except the one which says that every element have finite order .

how can we use this group to reach the asked group ?

share|improve this question
3  
Hint for 1: Consider an infinite number of copies of the same group. Hint for 2: Remember those restricted direct products. –  Tobias Kildetoft May 28 '13 at 19:34
add comment

4 Answers

up vote 2 down vote accepted

Very nice question!

You've got some answers for $(b)$. With respect to answering $(a)$, (which is, indeed, a challenging question):

I think you'll find the following post from MathOverflow very handy:

Is there a finitely generated nontrivial group $G$ such that $G\cong G\times G$?

It was asked by Martin Brandenburg, and there are two excellent answers (in the affirmative) given there.

share|improve this answer
    
Have you read the linked post, Maths Lover? –  amWhy Jun 9 '13 at 22:12
    
+1 $\frac{+}{0}$ –  B. S. Jun 22 '13 at 0:18
    
@BabakS. , this is nice , + infinty O_O :D –  Maths Lover Jun 22 '13 at 3:30
add comment

Hint for b: Consider a quotient of $\mathbb{Q}$.

share|improve this answer
add comment

I think for b you can consider $G=\mathbb Z(p^{\infty})$ where $p$ is a prime as well.

share|improve this answer
add comment

For the second problem, you can use the subgroup of $\mathbb{Z}_1\times \mathbb{Z}_2\times \cdots\times \mathbb{Z}_n \times \cdot$ consisting of all sequences $(a_1,a_2,a_3,\dots)$ such that all but finitely many of the $a_i$ are $0$.

share|improve this answer
    
This subgroup is the direct sum of the $\mathbb{Z}_n$, and is often denoted by $\mathbb{Z}_1 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_3 \oplus \ldots$ –  Mikko Korhonen May 29 '13 at 8:53
add comment

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.