The image contains a set of mathematical tasks and problems, primarily focused on group theory and related algebraic structures. Here's a breakdown of the main points:

1. Checking if an algebra is a group:

   • Verify the group axioms: closure, associativity, identity, and invertibility.
   • Check whether the group is Abelian (commutative).

2. Finding orders:

   • Determine the order (or size) of each element in the group.

3. Subgroups and subgroup lattice:

   • Identify all subgroups of the group.
   • Construct the subgroup lattice (a diagram showing inclusion relationships among subgroups).

4. Center and commutator subgroup:

   • Compute the center of the group (elements that commute with all other elements).
   • Determine the commutator subgroup (generated by all commutators $[a, b] = a^{-1}b^{-1}ab$).

5. Cosets and normal subgroups:

   • Find cosets of each subgroup.
   • Identify all normal subgroups of the group.

6. Properties of lattice of subgroups:

   • Determine if the subgroup lattice is modular, distributive, or complemented.
   • For a complemented lattice, find complements for all elements.

7. Roots of unity (Specific examples):

   • Problem 13.1 and 13.2 involve the set of all complex roots of $x^6 = 1$ and $x^8 = 1$ with multiplication.

Let me know which specific task you'd like me to work on or explain further!

5 Related Questions

1. How can we check if a set with a given operation forms an Abelian group?
2. What is the method for constructing subgroup lattices for small groups?
3. How do we compute the order of an element in a finite group?
4. What are the properties of the center and commutator subgroup, and why are they significant in group theory?
5. What does it mean for a lattice of subgroups to be modular or distributive?

Tip

For groups formed by roots of unity, remember that the order of an element corresponds to the smallest positive integer $n$ such that the element raised to $n$ equals 1. This concept simplifies the computation of orders.