UNDERSTANDING COSETS

Which of the following sets is a group under addition

• Integers

• Natural numbers

• Odd numbers

• Even numbers

Which of the following is a subgroup of ℤ under addition?

• {1, 2, 3}

• {0, 2, 4, 6, …}

• {3, 6, 9}

• {1, 3, 5}

The identity element for a group under multiplication mod 7 is:

• 0

• 1

• 7

• None of the following

In group theory, cosets help us understand:

• Only even numbers

• Polynomial equations

• Structure and partition of a group

• Matrices only

Let H = {1, 4} in ℤ₅ under multiplication. What is the coset 2H?

• {2, 3}

• {2, 4}

• {2, 1}

• {2, 3}

Let G = ℤ, H = multiples of 4. What is the coset 1 + H?

• {1, 5, 9, 13, …}

• {0, 4, 8, 12, …}

• {–1, –5, –9, …}

• {2, 6, 10, 14, …}

Let G = ℤ and H = 2ℤ. Which of the following is a left coset of H?

• {1, 3, 5, 7, …}

• {2, 4, 6, 8, …}

• {0, 1, 2, 3, …}

• {1, 2, 3, 4, …}

If G = ℤ and H = 3ℤ, what is 1 + H?

• {…, –4, –1, 2, 5, …}

• {…, –2, 1, 4, 7, …}

• {…, –3, 0, 3, 6, …}

• {…, –5, –2, 1, 4, 7, …}

Which of the following best describes a left coset of H in G?

• A subgroup of G

• The product of elements of H

• A set formed by multiplying an element of G on the left with each element of H

• A set of all elements in G

In ℤ₉ under addition, the subgroup generated by 3 is:

• {0, 3, 6}

• {3, 6, 9}

• {1, 4, 7}

• {0, 1, 2}