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Ques-1-Let-G-be-a-group-then-show-that-for-each-a-G-a-unique-element-e-G-a-e-e-a-a-Ques-2-If-a-G-x-G-and-x-is-unique-show-that-if-x-a-e-then-a-x-e-Hello-




Question Number 191986 by Mastermind last updated on 04/May/23
Ques. 1  Let (G,∗) be a group, then show  that for each a∈G, ∃ a unique   element e∈G ∣ a∗e=e∗a=a    Ques. 2  If a∈G ⇒ x∈G and x is unique  show that if x∗a=e, then a∗x=e.      Hello!
$$\mathrm{Ques}.\:\mathrm{1} \\ $$$$\mathrm{Let}\:\left(\mathrm{G},\ast\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{group},\:\mathrm{then}\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{for}\:\mathrm{each}\:\mathrm{a}\in\mathrm{G},\:\exists\:\mathrm{a}\:\mathrm{unique}\: \\ $$$$\mathrm{element}\:\mathrm{e}\in\mathrm{G}\:\mid\:\mathrm{a}\ast\mathrm{e}=\mathrm{e}\ast\mathrm{a}=\mathrm{a} \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{2} \\ $$$$\mathrm{If}\:\mathrm{a}\in\mathrm{G}\:\Rightarrow\:\mathrm{x}\in\mathrm{G}\:\mathrm{and}\:\mathrm{x}\:\mathrm{is}\:\mathrm{unique} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{x}\ast\mathrm{a}=\mathrm{e},\:\mathrm{then}\:\mathrm{a}\ast\mathrm{x}=\mathrm{e}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Hello}! \\ $$
Answered by AST last updated on 04/May/23
Ques.1 follows from the property of a group  that for every a∈G,∃ a unique element   e∈G∣a∗e=e∗a=a    2)  x∗a=e⇒x^(−1) ∗x∗a=x^(−1) ∗e⇒a=x^(−1) ∗e  ⇒a∗x=x^(−1) ∗e∗x=x^(−1) ∗x∗e=e
$${Ques}.\mathrm{1}\:{follows}\:{from}\:{the}\:{property}\:{of}\:{a}\:{group} \\ $$$${that}\:{for}\:{every}\:{a}\in{G},\exists\:{a}\:{unique}\:{element}\: \\ $$$${e}\in{G}\mid{a}\ast{e}={e}\ast{a}={a} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\:{x}\ast{a}={e}\Rightarrow{x}^{−\mathrm{1}} \ast{x}\ast{a}={x}^{−\mathrm{1}} \ast{e}\Rightarrow{a}={x}^{−\mathrm{1}} \ast{e} \\ $$$$\Rightarrow{a}\ast{x}={x}^{−\mathrm{1}} \ast{e}\ast{x}={x}^{−\mathrm{1}} \ast{x}\ast{e}={e} \\ $$
Commented by Mastermind last updated on 04/May/23
Thank you so much sir
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{sir} \\ $$

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