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# TIFR 2013 Problem 6 Solution - Finite order in Infinite Groups

TIFR 2013 Problem 6 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Contemporary Abstract Algebra by Joseph A. Gallian. This book is very useful for the preparation of TIFR Entrance.

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## Problem:True/False?

Every infinite abelian group has at least one element of infinite order.

## Hint:

Characteristic of a ring.

## Discussion:

What if we have an element of finite order, and attach it to infinitely many "inert" elements?

Consider $\mathbb{Z_2}$. ( Note that,   $\mathbb{Z_n}$ is an integral domain iff n is prime.  And a finite integral domain is a field. Therefore, $\mathbb{Z_2}$ is a (finite) field. )

$\mathbb{Z_2}[x]$ is the ring of polynomials with coefficients from $\mathbb{Z_2}$. A ring is a group with some extra conditions. $\mathbb{Z_2}[x]$ is a group with respect to addition. What is the order of each element in this group?

Let $p(x)=a_0+a_1x+a_2x^2+...a_nx^n\in\mathbb{Z_2}[x]$

Since $a_0,a_1,...,a_n\in\mathbb{Z_2}$ we have $2a_0,2a_1,...,2a_n=0$. Note that 0 here means the identity element of the group $\mathbb{Z_2})$, not the 0 we use in Real number system.

Hence, $p(x)+p(x)=2p(x)=0$. Here 0 is the identity of $\mathbb{Z_2}[x]$.

Therefore, each element of the group  $(\mathbb{Z_2}[x],+)$ has order 1 or 2.

The elements $(1,x,x^2,x^3,...)$ are all distinct members of this group, hence $(\mathbb{Z_2}[x],+)$ is an infinite group, where every element has finite order.

Isomorphic examples: The set of all sequence with elements from $\mathbb{Z_2}$ & the infinite Cartesian product $\mathbb{Z_2}\times \mathbb{Z_2}\times\mathbb{Z_2}\times...$ are two examples which follows.

Non-isomorphic  simple generalizations: Of course, there is nothing special about the number 2. We can take any prime (natural) number. Can we take any natural number? Although in courses of algebra we usually deal with $\mathbb{Z_p}$, p-prime, here there is no need to restrict ourselves to prime p-s only. Indeed we can take any $\mathbb{Z_n}[x]$, with $n\in\mathbb{N}$. It may not be 'good enough' as a ring ($\mathbb{Z_n}$ not being integral domain for n not prime and all that...) but it is sufficient for our example.

## Chatuspathi

• What is this topic: Abstract Algebra
• What are some of the associated concept: Finite Order,Characteristic of Ring
• Book Suggestions: Contemporary Abstract Algebra by Joseph A. Gallian