Alg7.5

From Exampleproblems

Jump to: navigation, search

Prove that if $G\,$ is a group and $H, K\,$ are subgroups of $G\,$ , then $H\cap K\,$ is a group.

I prove it with the subgroup criteria:

If $\empty \ne H\subseteq G\,$ and $\forall a,b\isin H, ab^{-1}\isin H\,$ , then $H\,$ is a subgroup.

First I prove that $1_G = 1_H = 1_K\,$ which implies $H\cap K \ne \empty\,$ .

$\forall a\isin G, 1_G a=a\,$ and $\forall b\isin H, 1_H b = b\,$ . Since $1_H,b\isin G, 1_H b=b\implies 1_H = 1_G\,$ . Similarly for $1_K\,$ . Since $1_G=1_H=1_K, 1_G\isin H\cap K\ne\empty\,$ .

Second, I prove that $\forall a, b\isin H\cap K, ab^{-1}\isin H\cap K\,$ .

If $a,b\isin H\cap K\,$ then $a,b\isin H\,$ and $a,b\isin K\,$ . This implies $ab^{-1}\isin H\,$ and $ab^{-1}\isin K\,$ since $H\,$ and $K\,$ are both subgroups of $G\,$ , so $ab^{-1}\isin H\cap K\,$ . Therefore, $H\cap K\,$ is a group.

Abstract Algebra

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Alg7.5"

Views

Personal tools

Create an account or log in

Search

Toolbox

Argan Oil
Natural Skin Care
Organic Skin Care