Question

Let $G$be an abelian group and $T$the set of elements of finite order in $G$Prove that Every nonzero element of the quotient group $G/T$ has infinite order.

Short answer

It is proved that, every non-zero element of the quotient group $G/T$has infinite order.

Step-by-step solution

Step 1:Group G and T

Given that $G$ is an abelian group and $T$ is the set of elements of finite order in $G$.

G/T has infinite order

Let $\left[a\right]\in G/T$is a non-zero element.

Then,$a\notin T$and $\left[a\right]$has finite order $m.$

The element $[a{]}^m$can be written as:

$[a{]}^m=\left[a^m\right]$

$=\left[a^{\left|a\right|}\right]$

$=e$

Thus, $[a{]}^m=e$implies $a^m\in T$has finite order$n$that is $a^{mn}=e$

Hence, $a\in T$is a contradiction to $a\notin T$

Therefore, every non-zero element of the quotient group $G/T$has infinite order.

.