**Given that πΊ = {π β β|π β β1} and π β π = π + π + ππ, show that (πΊ, β) is indeed a group.**

The **Answer to the Question**

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**Here's the Solution to this Question**

group properties:

associativity:

$(a*b)*c=(π + π + ππ)*c=π + π + ππ+c+(π + π + ππ)c=$

$=π + π + ππ+c+ac+bc+abc$

$a*(b*c)=a*(b+c+bc)=a+b+c+bc+a (b+c+bc)=$

$=a+b+c+bc+ab+ac+abc$

$(a*b)*c=a*(b*c)$

unit element e:

$e*a=e+a+ae=a$

$e=0$

inverse element b=a-1:

$a*b=a+b+ab=e=0$

$b=-\frac{a}{a+1}$

so, (πΊ, β) is a group