Proof about identity element of a group

hoopsmax25
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Homework Statement


If G is a group, a is in G, and a*b=b for some b in G (* is a certain operation), prove that a is the identity element of G


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The Attempt at a Solution


I feel like you should assume a is not the identity element and eventually show that a= the identity. but I am not sure how to show that.
 
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Hint: Think about using ##b^{-1}##.
 
I thought about using the inverse of b but I'm not sure how to plug it in?
 
You don't "plug it in". You could try *-ing things with it.
 
Could you do a*b*b-inverse=b*b-inverse? and go from there?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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