Hints for Homework #3
Chapter 2, #34
The identity of this group is the standard 3-by-3
identity matrix. To determine the inverse of
an arbitrary matrix in the Heisenberg group,
consider the product formula given and determine
relations on a', b', c' in terms of the a, b, c
that make the product equal to the identity matrix.
Your inverse should be two-sided. So when you
compute your a', b', c', multiply the matrices
in the opposite order to get a check on your answer.
Chapter 2, #35
Remember that the Euclidean algorithm finds you the
multiplicative inverse of a modulo b if a and b
are two relatively prime integers.
Chapter 3, #6
To show x^4 <> e, you could assume that x^4 = e and try
to get a contradiction. If x^4 = e and x^6 = e, what
does this imply about x^2 ?
Chapter 3, #22
Make sure to give a justification in addition to a yes/no
answer. Start by writing down the definition of the center
of a group. Choose a, b to be arbitrary elements of the
center and try to show that ab = ba.
Chapter 4, #14
Read the fundamental theorem on cyclic groups and try to
understand the relationship between divisibility of orders
and subgroups of a cyclic group. If you can't see how to
proceed, consider the following analogous problem:
Let n be a positive integer divisible by exactly three
distinct integers: 1, 7, and n itself. What is n?
Chapter 4, #17
Contributed by Michelle Yakaboski
Question:
Complete the following statement: |a|=|a^2| if and only if |a| is...
Hint:
Using theorem 4.2 we know < a^k > = < a^gcd(n,k)> and |a^k| = n/gcd(n,k).
Then find the order for k=2 and for k=1.
You will see that n/gcd(n,2) = n.
Then set the order of |a^2| = |a|.