Alg7.6

From Exampleproblems

Prove that $\empty \ne H\subseteq G\,$ and $\forall a,b\isin H, ab^{-1}\isin H\,$ , if and only if $H\,$ is a subgroup of $G\,$ .

If $H\,$ is a subgroup of $G\,$ , by the properties of identity, closure and inverse, it is true that $\empty\ne H\,$ and $\forall a,b\isin H, ab^{-1}\isin H\,$ .

If $\forall a,b\isin H\ne\empty, ab^{-1} \isin H\,$ , show that this implies $H\,$ is a subgroup of $G\,$ .

Identity: First let $b=a\,$ so that $aa^{-1}=e\isin H\,$ .

Inverse: Let $a=e\,$ so that $eb^{-1}=b^{-1}\isin H\,$ .

Closure: Let $b=b^{-1}\,$ so that $a(b^{-1})^{-1}=ab\isin H\,$ .

Associativity: If $\exists a,b,c \isin H \mid (ab)c \ne a(bc) \,$ , then $\exists a,b,c \isin G \mid (ab)c \ne a(bc) \,$ which is not true by the assumption that $G\,$ is a group.

Abstract Algebra

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