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UPSC 2013 Maths Optional Paper 2 Q1b — Step-by-Step Solution

10 marks · Section A

Cyclic groups · Algebra · asked 8× in 14 yrs · Read the full method →

Question

Give an example of an infinite group in which every element has finite order.

Technique

Standard counterexample to “infinite group $\Rightarrow$ has element of infinite order” — both $\mathbb{Q}/\mathbb{Z}$ and roots of unity work.

Solution

Example. $G=\mathbb{Q}/\mathbb{Z}$ — the additive group of rationals modulo integers.

Step 1 — $G$ is an infinite group

Elements of $G$ are cosets $r+\mathbb{Z}$ with $r\in\mathbb{Q}$. Two cosets $r+\mathbb{Z}$ and $s+\mathbb{Z}$ are equal iff $r-s\in\mathbb{Z}$. So a canonical representative is $r\in[0,1)\cap\mathbb{Q}$.

The set $[0,1)\cap\mathbb{Q}$ is infinite (contains $1/n$ for each $n\ge 2$), so $G$ is infinite.

Step 2 — Every element has finite order

Let $g=r+\mathbb{Z}\in G$, written as $r=p/q$ in lowest terms with $p,q\in\mathbb{Z},\;q\ge 1$. Then

since $p\in\mathbb{Z}$. So the order of $g$ divides $q$ — finite (specifically, equal to $q/\gcd(p,q)=q$ for $p/q$ in lowest terms with $\gcd(p,q)=1$).

Answer