TSSFL TECHNOLOGY STACK

Posted: **Fri Jan 15, 2016 11:08 am**

$H$ is a subgroup of a group $G$ if the identity element belongs to $H$ ; $a, b \in H \implies \ ab\in H$ and also the inverse element belongs to $H$ . We illustrate this by example.

Example 1

If $G$ is an abelian group and if $H = \{a\in G: \ a^2 = e \},$ show that $H$ is a subgroup of $G$ .

Solution

(i) $a^2 = e,$ hence $e\in H$

(ii) If $a, \ b \in H,$ then $(ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = (aa)(bb) = a^2b^2 = ee = e,$ so $ab \in H$

(iii) If $a \in H,$ that's $a^2 = e$ , multiplying both sides of the equality by $(a^{-1})^2,$ we get

$a^2(a^{-1})^2 = e(a^{-1})^2 \\ \\ (aa)(a^{-1}a^{-1}) = (a^{-1})^2 \\ \\ a(aa^{-1})a^{-1} = (a^{-1})^2 \\ \\ aa^{-1} = (a^{-1})^2 \\ \\ e = (a^{-1})^2.$

Hence $a^{-1}\in H$ and thus $H$ is a subgroup of $G$

TSSFL TECHNOLOGY STACK

Subgroups - Solved Example

Subgroups - Solved Example