1. Suppose G = ?a? is a cyclic group of order 16. List all...

1. Suppose G = ?a? is a cyclic group of order 16. List all distinct subgroups of G and their elements. 2. Is U(12) a cyclic group? If so, prove so. If not, demonstrate that it is not. 3. Let f: G ? H be an isomorphism of groups. Show that if g generates G then f(g) generates H.

Transcript

00:01 I hear for the given question in the first part.

00:04 We are given that suppose g is a cyclic group of order 16.

00:15 Then a cyclic group is generated by a single element a then here we need to find the distinct subgroup of g.

00:32 We can correspond the power of a since the order of g is 16.

00:37 The highest power will be 15 e to the power 15 will be highest as we are given order is equal to 16.

00:48 So here we can list the different subgroup as here list of subgroups will be tribal subgroup, which is containing identity element.

01:11 Whereas the subgroup generated by e to the power 8, which is e a to the power 8 now similarly generated by a to the power 4 e e to the power 4 e to the power 8 e to the power 12 similarly generated by a square which is e a square comma a to the power 4 e to the power 6 e to the power 8 comma e to the power 10 e to the power 12 e to the power 14 now further subgroup containing whole group g itself, which is a a square up to a to the power 16.

01:56 So here now as we can observe that here for each subgroup, if we divide the order of g as required by the lagrange's theorem, so here these are the list of the subgroup and this is the solution of the first part now here for the second part.

02:14 We are given that the group of 12 consists of integer less than 12 integer less than 12, but they are relative prime.

02:36 So here in our case, we need to determine if this is a cyclic group.

02:40 So here if you 12 is cyclic group, then we need to find element or who generates the entire group.

02:49 So we need to find the generator of entire group.

02:56 So here in our case, we know that here the list of element for you 12 is equal to 1 5 7 and 11 now here as we can observe that here the value of 1 square is equal to 1 further 5 square is equal to 5 similarly 7 square is equal to 1 and 11 square is equal to 1.

03:22 So here each group has like we can say that each element is 2 and there is no element whose power can generate the entire group...

Question

1. Suppose G = ?a? is a cyclic group of order 16. List all distinct subgroups of G and their elements. 2. Is U(12) a cyclic group? If so, prove so. If not, demonstrate that it is not. 3. Let f: G ? H be an isomorphism of groups. Show that if g generates G then f(g) generates H.

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