Some Problems with Hints...

Many of the students are asking their doubts by sending a picture in whatsapp. Here I post the Hints for the answers, so that others also can use the problems. As I have too little time to type all in Latex, here simply I post the photo. 

Any more suggestion, comments are welcome.

Also I post Problems and Hints, that I found in some groups asked and solved by others. On behalf of myself and readers of this post, I express my sincere thanks to them. 

*Problems will be updated regularly*

Warning: ALL THE ANSWERS AND HINTS NEED  NOT BE CORRECT. VERIFY THOROUGHLY.  ALSO IF YOU FIND ANY MISTAKES PLEASE WARN ME. I TRY TO CORRECT. 

Hint: https://math.stackexchange.com/questions/697738/dimension-of-the-vector-space-of-homogeneous-polynomials

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Hint: Use M-l inequality in complex integration. Also Here the question is incorrect. Instead of |z|=1, take |z|=2.

https://math.stackexchange.com/questions/666229/what-is-ml-inequality-property-of-complex-integral

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Hint: Use binomial expansion for numerator.  find the coefficient of x^2n.  From that find the coefficient of 1/z in the series of given function

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Hint: A similar problem: https://math.stackexchange.com/questions/811391/f-nx-nx1-xn-determine-whether-the-sequence-f-n-converges-uniformly-o

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See Question 11

 Solution:

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If both graph G and its complements are trees, then the number of vertices is ________

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Hint: Winding number https://en.wikipedia.org/wiki/Winding_number

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Number of automorphisms of Z, Zn, Sn and Klien four group.

Hint: A automorphism of a cyclic group is completely determined by the image of a generator of the cyclic group.

In Z, the generators are $\pm 1$. So by finding the image of 1, the problem can be solved. Therefore two automorphisms, namely $1 \mapsto 1$ and $1 \mapsto -1$ .

In Zn, consider the generator 1. suppose totally m(=???) generators we have, then 1 can be mapped to m values.

But Sn and  Klien four group are not cyclic. Anyway here we can do a similar work by taking a "generator set". In Sn two elements generate the group, namely a 2-cycle $\tau$ and a 3-cycle $\sigma$. Keeping in mind the fact "Automorphism preservs orders" here too we can find the number of ways to map  $\tau$ and $\sigma$.

The case of Klein four group is left to the reader.

https://math.stackexchange.com/questions/103390/follow-up-to-question-autg-for-g-klein-4-group-is-isomorphic-to-s-3

https://math.stackexchange.com/questions/1538681/finding-the-automorphisms-of-s-3-by-looking-at-the-orders-of-the-elements