202101141501 Homework 2 (Q1)

Let $H$ be a subset of the group $G$ . Show that $H$ is a subgroup of $G$ if and only if $H\neq \emptyset$ and $gh^{-1}\in H$ whenever $g,h\in H$ .

Attempts.

(only-if part) If $H$ is a subgroup of $G$ , then $H$ must contain the identity $e$ and both $x^{-1}$ and $xy$ is inside $H$ whenever $x,y\in H$ . Thus $H\neq \emptyset$ and for any $g,h\in H$ , $h^{-1}$ is inside $H$ . Thus $gh^{-1}\in H$ .

(if-part) First, because $H\neq \emptyset$ , there exists some $x$ in $H$ such that $e=xx^{-1}\in H$ (by the condition $gh^{-1}\in H$ whenever $g,h\in H$ ). This proves the existence of the identity. Secondly, having proven $e$ is in $H$ , and noted that $x^{-1}=ex^{-1}\in H$ whenever $e,x\in H$ , one proves the existence of the inverse. Thirdly, $x,y\in H$ implies $xy\in H$ because $y\in H\Rightarrow y^{-1}\in H$ and also $xy=x(y^{-1})^{-1}$ . $H$ is therefore a subgroup of $G$ .

M	T	W	T	F	S	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31

Leave a Reply Cancel reply