Is Munkres Lemma 81.1 Misinterpreted Regarding Normal Subgroups?

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[SOLVED] munkres lemma 81.1

Homework Statement


Please stop reading this if you do not have Munkres.

The statement of this lemma implies that pi_1(B,b_0)/H_0 is a group. That does not make sense to me because H_0 is not necessarily normal in pi_1(B,b_0).

Homework Equations





The Attempt at a Solution

 
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Replace the word subgroup with subset and the statement becomes less confusing. Munkres is only talking about \pi_1(B, b_0) / H_0 as a set of cosets.
 

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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