Let H be a non−empty subset of a group G, prove that the follow− ing are equivalent 1) H is a subgroup of G 2) for a,b ∈ H, ab^(−1) ∈ H 3) for a,b ∈ ab ∈ H 4) for a ∈ H, a^(−1) ∈ H Hint: prove 1)→2)→3)→4)→1) Help!!!


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Question Number 192077 by Mastermind last updated on 07/May/23

$Let H be a non - empty subset of$ $a group G, prove that the follow -$ $ing are equivalent$ $1) H is a subgroup of G$ $2) for a, b \in H, {ab}^{- 1} \in H$ $3) for a, b \in ab \in H$ $4) for a \in H, a^{- 1} \in H$ $H i n t : prove 1) \to 2) \to 3) \to 4) \to 1)$ $Help!!!$
	Answered by deleteduser1 last updated on 07/May/23

	$H i s a s u b g r o u p o f G \Rightarrow H i s a g r o u p w i t h e l e m e n t s$ $f r o m G .$ $\Rightarrow a b \in H f o r a, b \in H$ $S i n c e b \in H, b^{- 1} \in H$ $H e n c e, s i n c e a, b^{- 1} \in H, {a b}^{- 1} \in H$ $\Rightarrow 1) \Rightarrow 3) \Rightarrow 2) \Rightarrow 4)$
	Commented by Mastermind last updated on 07/May/23

	$Thank you so much sir$
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