Please help me with these abstract algebra questions.

Project - MTH 4050

The group I have been working with: Group: ????4[i]= {a+bi | a,b ????4}, addition of complex numbers.

1. Categorize the elements in your group using element order. What are the possible element orders? Which elements have the same order? How many elements are there of each order?

(This is what I have already done... Does it make sense?)

For the group ????4[i], there are three possible element orders: order 1, order 2, and order 4. The order of the element 0+0i has an order of 1. This element is the only element of the group that has order 1, making it the identity element. Another possible element order is order 2. There are three elements within the group that share the same order 2: 2+0i, 0+2i, and 2+2i. The third possible element order is an order of 4. There are twelve elements within the group that share the same order 4: 1+0i, 3+0i, 0+1i, 0+3i, 1+1i, 1+2i, 1+3i, 2+1i, 2+3i, 3+1i, 3+2i, and 3+3i.

2. Review your results from Homework #2 and Homework #3. In those assignments, you explored subgroups of the original group. That gave you some sense of the overall structure of your group, and you started to see some underlying substructures. Review those past assignments to see if you can identify any other structural characteristics of this group.

3. Look for groups that are structurally the same as the group corresponding to your focal point from Homework #3. Many of you chose a focal point that resembles a dihedral group or a subgroup of a dihedral group. What other groups are structurally similar to or different from your focal point group?

4. Look for subgroups of the group corresponding to your focal point from Homework #3. What smaller patterns can you identify in the larger pattern?

5. Start thinking about how your group can be viewed as the external direct product of two or more smaller groups. What can you say about those smaller groups? How do they help you understand the structure of the larger group you and your teammates looked at in the homework sets?

Links to Homework #2: https://docs.google.com/document/d/1fPBaryuXhnSeBpzM4iFbwo0eTYtNnoLWAiUbfv2Yvls/edit?usp=sharing

And Homework #3:

https://docs.google.com/document/d/1Kh2ds6VBDnBruDwvBetbZBsuqTdKNKn-qQLsR8XW0f4/edit?usp=sharing