when you put your question in "Quote", we can't quote you anymore.
Question 1: suppose is the unique element of order 2 and is any element of the group. then has order 2 and thus
Question 2: proof by induction on if then every non-identity element of G has order p and we're done. let and suppose that the claim is true for any group of order
if G is cyclic, then there's nothing to prove. if it's not cyclic, then choose now apply induction on
Question 3: the center has only one element, i.e. the identity matrix. suppose G is the group of all unit upper triangular matrices over a field or a unitary commutative ring. as usual
let be the matrix with 1 in its (i,j)-entry and 0 elsewhere. let we have for some scalars now let then and so we must
have which gives us: using the fact that we'll have: and this gives us:
therefore and thus: