Question

Let G be an abelian group and n a fixed positive integer.

(a) Prove that $H=\left\{a\in G\overline{)a^n=e}\right\}$is a subgroup of G.

Short answer

Answer

Itis proved that $H=\left\{a\in G\overline{)a^n=e}\right\}$is a subgroup of G.

Step-by-step solution

Step-by-Step Solution Step 1: Subgroup Theorem

A non-empty subset H of a group G is a subgroup of the latter if the elements a and b belong to H, then the element ab belongs to H, and if the element a belongs to H , the element a^-1belongs to H.

Show that H=a∈Gan = eis a subgroup of G

Consider the following set:

$H=\left\{a\in G\overline{)a^n=e}\right\}$

Consider two elements and in the set with the following properties:

aⁿ = e

And,

bⁿ= e

Find the product of above two elements:

$\begin{array}{c}{\left(ab\right)}^n=\left(ab\right)\left(ab\right)\left(ab\right)\dots \left(ab\right)\\ =\left(aaa\right)\left(bbb\right)\\ =a^n{b}^n\\ =e\end{array}$

Therefore, the product ab is present in the set H.

Find the inverse of element a as follows:

$\begin{array}{c}{\left(a^{-1}\right)}^n={\left(a^n\right)}^{-1}\\ =e^{-1}\\ =e\end{array}$

Therefore, the inverse of element a is also present in the set H. Hence, set H is a subgroup G.

Therefore, it is proved that $H=\left\{a\in G\left|a^n=e\right.\right\}$is a subgroup of G.