Hello MHF,

Now I finished reading Artin, Chapter 2 and was attempting exercises. Few problems look really tricky. I will be grateful if anybody could help me out with these:

1) Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.2) Prove that a group whose order is a power of prime p,contains an element of order p.

:If denotes a general element in G, I proceeded like this:Something that looked like a promising direction

Lets say there does not exist an element of order prime. Consider the set where I have not considered the identity. Then it forces S to be a permutation of G \ {e}. But I could not proceed further...

Clearly it has the set of all scalar matrices(matrices of form cI) as its subgroup... But do you know any other matrix that commutes with all unit upper triangular matrices?3) What is the center of the group of all unit upper triangular matrices?

I would appreciate hints compared to solutions.

Thanks,

Srikanth