Question

Find all subgroups of G with multiplication modulo 15.

a) {1,4,11,14} and {1,11}
b) {1,2,4,7,8,11,13,14} and {1,11}
c) {1,4,7,11,14} and {1,11}
d) {1,2,4,7,8,11,13,14} and {1,4,11,14}

Answer 1

To find all subgroups of G with multiplication modulo 15, we need to analyze the given sets and determine if they satisfy the properties of a subgroup.

Step-by-step explanation:

In order to find all the subgroups of G with multiplication modulo 15, we need to determine which subsets of G satisfy the properties of subgroups. A subgroup is a subset of G that is closed under the group operation, contains the identity element, and contains the inverses of its elements.

Using this information, we can analyze the given sets:

• a) Set {1,4,11,14} satisfies the properties of a subgroup because it is closed under multiplication modulo 15, contains the identity element (1), and contains the inverses of its elements.
• b) Set {1,2,4,7,8,11,13,14} does not satisfy the properties of a subgroup because it is not closed under multiplication modulo 15.
• c) Set {1,4,7,11,14} does not satisfy the properties of a subgroup because it does not contain the inverses of all its elements.
• d) Set {1,2,4,7,8,11,13,14} satisfies the properties of a subgroup because it is closed under multiplication modulo 15, contains the identity element (1), and contains the inverses of its elements.