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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

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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 ∈ 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!!!
$$\mathrm{Let}\:\mathrm{H}\:\mathrm{be}\:\mathrm{a}\:\mathrm{non}−\mathrm{empty}\:\mathrm{subset}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{group}\:\mathrm{G},\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{follow}− \\ $$$$\mathrm{ing}\:\mathrm{are}\:\mathrm{equivalent} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{H}\:\mathrm{is}\:\mathrm{a}\:\mathrm{subgroup}\:\mathrm{of}\:\mathrm{G} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{for}\:\mathrm{a},\mathrm{b}\:\in\:\mathrm{H},\:\mathrm{ab}^{−\mathrm{1}} \:\in\:\mathrm{H} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{for}\:\mathrm{a},\mathrm{b}\:\in\:\mathrm{ab}\:\in\:\mathrm{H} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{for}\:\mathrm{a}\:\in\:\mathrm{H},\:\mathrm{a}^{−\mathrm{1}} \:\in\:\mathrm{H} \\ $$$$ \\ $$$$\left.\mathrm{H}\left.\mathrm{i}\left.\mathrm{n}\left.\mathrm{t}\left.:\:\mathrm{prove}\:\mathrm{1}\right)\rightarrow\mathrm{2}\right)\rightarrow\mathrm{3}\right)\rightarrow\mathrm{4}\right)\rightarrow\mathrm{1}\right) \\ $$$$ \\ $$$$\mathrm{Help}!!! \\ $$
Answered by AST last updated on 07/May/23
H is a subgroup of G⇒H is a group with elements from G. ⇒ab∈H for a,b∈H Since b∈H,b^(−1) ∈H Hence,since a, b^(−1) ∈H, ab^(−1) ∈H ⇒ 1)⇒3)⇒2)⇒4)
$${H}\:{is}\:{a}\:{subgroup}\:{of}\:{G}\Rightarrow{H}\:{is}\:{a}\:{group}\:{with}\:{elements} \\ $$$${from}\:{G}. \\ $$$$\Rightarrow{ab}\in{H}\:{for}\:{a},{b}\in{H} \\ $$$${Since}\:{b}\in{H},{b}^{−\mathrm{1}} \in{H} \\ $$$${Hence},{since}\:{a},\:{b}^{−\mathrm{1}} \:\in{H},\:{ab}^{−\mathrm{1}} \in{H} \\ $$$$ \\ $$$$\left.\Rightarrow\left.\:\left.\mathrm{1}\left.\right)\Rightarrow\mathrm{3}\right)\Rightarrow\mathrm{2}\right)\Rightarrow\mathrm{4}\right) \\ $$
Commented by Mastermind last updated on 07/May/23
Thank you so much sir
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{sir} \\ $$

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