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Question-76328




Question Number 76328 by arkanmath7@gmail.com last updated on 26/Dec/19
Commented by arkanmath7@gmail.com last updated on 26/Dec/19
sol. plz
$${sol}.\:{plz} \\ $$
Commented by kaivan.ahmadi last updated on 26/Dec/19
1) aa^(−1) =1∈H⇒a≡a^(−1)   2)if a≡b⇒ab^(−1) ∈H⇒  ba^(−1) ab^(−1) =1∈H⇒ba^(−1) ∈H⇒b≡a  3)if a≡b and b≡c⇒ab^(−1) ∈H and bc^(−1) ∈H⇒    ac^(−1) =ab^(−1) bc^(−1) ∈H⇒a≡c
$$\left.\mathrm{1}\right)\:{aa}^{−\mathrm{1}} =\mathrm{1}\in{H}\Rightarrow{a}\equiv{a}^{−\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){if}\:{a}\equiv{b}\Rightarrow{ab}^{−\mathrm{1}} \in{H}\Rightarrow \\ $$$${ba}^{−\mathrm{1}} {ab}^{−\mathrm{1}} =\mathrm{1}\in{H}\Rightarrow{ba}^{−\mathrm{1}} \in{H}\Rightarrow{b}\equiv{a} \\ $$$$\left.\mathrm{3}\right){if}\:{a}\equiv{b}\:{and}\:{b}\equiv{c}\Rightarrow{ab}^{−\mathrm{1}} \in{H}\:{and}\:{bc}^{−\mathrm{1}} \in{H}\Rightarrow \\ $$$$ \\ $$$${ac}^{−\mathrm{1}} ={ab}^{−\mathrm{1}} {bc}^{−\mathrm{1}} \in{H}\Rightarrow{a}\equiv{c} \\ $$$$ \\ $$

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