### Assignment #3

##### Due: Mar 13th, 2003 at 11am MST

1. At one point I claimed in class that every row and column of a group's "multiplication table" was a permutation of the group elements. Please prove this property.

2. At one point we discussed if the property in question #1 was also a sufficient condition for building groups. In other words, if we draw an n by n multiplication table where each row and each column is a permutation of the group elements, does that imply that this table represents a group?
(Hint: The answer is no; prove this by providing a counterexample.)

3. Prove that a finite group of even order always has some element i different from the identity, where i is its own inverse.

4. Consider the finite field GF(16) with irreducible polynomial m(x) = x4 + x3 + 1. We will, as usual, represent group elements as either polynomials in x, or hex number, or binary strings, with the usual mapping between these representations.
(a) What polynomial corresponds to the hex number 'a'?
(b) What is 'a' + '7' in this field?
(c) What is 'a' * '7' in this field?
(d) What is the additive inverse of 'e' in this field?
(e) What is the multiplicative inverse of 'e' in this field? (Please use the xtime() method and show your work.)