[Math] Subgroup criterion.

group-theory

I've been reading some stuff about algebra in my free time, and I think I understand most of the stuff but I'm having trouble with the exercises. Specifically, the following:

Prove that a nonempty subset $H$ of a group $G$ is a subgroup if for
all $x, y \in H$, the element $xy^{-1}$ is also in H.

Proving that the identity is in $H$ is easy: just take $x=y$, so $x x^{-1} = 1 \in H$. However, I'm having trouble showing that multiplication is closed and that each element in $H$ has an inverse. Can anyone give some hints?

Best Answer

  1. by given condition for any $x\in H$ we have $xx^{-1}=e$ is in $H$, denote identity element by $e$
  2. take any $x\in H$ and $e\in H$ so by the given condition $ex^{-1}=x^{-1}\in H$ so every element of $H$ has inverse in $H$.
  3. take any $x,y\in H$ as $y^{-1}\in H$ so by given condition $x(y^{-1})^{-1}=xy\in H$, which proves the closure property.
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