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Question Number 191986 by Mastermind last updated on 04/May/23

$$\mathrm{Ques}.1$$$$\mathrm{Let}(\mathrm{G},\ast )\mathrm{be}\mathrm{a}\mathrm{group},\mathrm{then}\mathrm{show}$$$$\mathrm{that}\mathrm{for}\mathrm{each}\mathrm{a}\in \mathrm{G},\mathrm{\exists }\mathrm{a}\mathrm{unique}$$$$\mathrm{element}\mathrm{e}\in \mathrm{G}\mid \mathrm{a}\ast \mathrm{e}=\mathrm{e}\ast \mathrm{a}=\mathrm{a}$$$$$$$$\mathrm{Ques}.2$$$$\mathrm{If}\mathrm{a}\in \mathrm{G}\Rightarrow \mathrm{x}\in \mathrm{G}\mathrm{and}\mathrm{x}\mathrm{is}\mathrm{unique}$$$$\mathrm{show}\mathrm{that}\mathrm{if}\mathrm{x}\ast \mathrm{a}=\mathrm{e},\mathrm{then}\mathrm{a}\ast \mathrm{x}=\mathrm{e}.$$$$$$$$$$$$\mathrm{Hello}!$$

Answered by deleteduser1 last updated on 04/May/23

$$Ques.1followsfromthepropertyofagroup$$$$thatforeverya\in G,\mathrm{\exists }auniqueelement$$$$e\in G\mid a\ast e=e\ast a=a$$$$$$$$2)x\ast a=e\Rightarrow x^{-1}\ast x\ast a=x^{-1}\ast e\Rightarrow a=x^{-1}\ast e$$$$\Rightarrow a\ast x=x^{-1}\ast e\ast x=x^{-1}\ast x\ast e=e$$

Commented by Mastermind last updated on 04/May/23

$$\mathrm{Thank}\mathrm{you}\mathrm{so}\mathrm{much}\mathrm{sir}$$

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