Exercises 6.5 Exercises
1.
Suppose that is a finite group with an element of order and an element of order Why must
2.
Suppose that is a finite group with elements. What are the orders of possible subgroups of
3.
Prove or disprove: Every subgroup of the integers has finite index.
4.
Prove or disprove: Every subgroup of the integers has finite order.
5.
List the left and right cosets of the subgroups in each of the following.
in
in
in
in
in
in
in
in
6.
Describe the left cosets of in What is the index of in
7.
Verify Euler's Theorem for and
8.
Use Fermat's Little Theorem to show that if is prime, there is no solution to the equation
9.
Show that the integers have infinite index in the additive group of rational numbers.
10.
Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.
11.
Let be a subgroup of a group and suppose that Prove that the following conditions are equivalent.
12.
If for all and show that right cosets are identical to left cosets. That is, show that for all
13.
What fails in the proof of Theorem 6.8 if is defined by
14.
Suppose that Show that the order of divides
15.
The cycle structure of a permutation is defined as the unordered list of the sizes of the cycles in the cycle decomposition For example, the permutation has cycle structure which can also be written as
Show that any two permutations have the same cycle structure if and only if there exists a permutation such that If for some then and are conjugate.
16.
If prove that the number of elements of order is odd. Use this result to show that must contain a subgroup of order 2.
17.
Suppose that If and are not in show that
18.
If prove that
19.
Let and be subgroups of a group Prove that is a coset of in
20.
Let and be subgroups of a group Define a relation on by if there exists an and a such that Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of in
21.
Let be a cyclic group of order Show that there are exactly generators for
22.
Let where are distinct primes. Prove that
23.
Show that
for all positive integers