How Do You Prove the Center of a Group is Nontrivial in Group Theory?

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Homework Statement


Let G be a group of order p^n where p is a prime number and n>o,
1.prove that the center Z(G) of G is nontrivial.
2.Suppose n=2. Prove that G is an abelian group


Homework Equations


i know what center is...


The Attempt at a Solution


help!
Z(G)=center=(in this case)centralizer of G?
but how to use?
It seems like i have the knowledge but do not know how to aplpy...
i have no idea for part 1 so cannot do part 2 as well...
 
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You might want to think about the conjugacy classes
 

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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