Math 366 Assignment 1
Due Monday, October 18
1. For each of the following sets/operations, provide a reason why the set/operation
does not form a group.
(a) The even integers with multiplication.
(b) Z4 with multiplication (mod four).
(c) Q (the rationals) with division.
2. Write out the Cayley (multiplication) table for U (10) = {1, 3, 7, 9} under
multiplication (mod 10). What is the inverse of each element?
3. An element x of a group is called idempotent if x2 = x. Prove that the
only idempotent element of a group is the identity element e.
4. Determine the orders of the following elements in the given groups:
(a) 2 ∈ (Z10 , +).
(b) 2 ∈ (Z×
17 , ∗).
(c) 3 ∈ (U (20), ∗) (U (20) is the set of all numbers less than 20 that are
relatively prime to 20).
5. (a) In some group G, suppose the inverse of a is b. Prove that b cannot
be the inverse of any other element of G.
(b) Suppose some group G has order 82. Why must there be at least one
non-identity element of G that is its own inverse?
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