Center (group theory) facts for kids

In abstract algebra, a branch of mathematics that studies groups, the center of a group is a special collection of elements. Imagine a group as a team where everyone has a specific role. The center of the group is like the small group of team members who can do their job in any order with anyone else on the team, and it won't change the outcome.

This collection of elements is called Z(G), where G stands for the group. The letter 'Z' comes from the German word Zentrum, which means 'center'.

So, if you have an element z in the center of a group G, it means that z can be combined with any other element g in the group, and the result will be the same whether you do z first then g, or g first then z. In math language, this is written as:

zg = gz

The center of a group is always a normal subgroup. This means it's a special kind of subgroup that has certain properties, making it very important in understanding the group's structure.

• A group is called abelian if its center is the entire group itself. This means every element in an abelian group commutes with every other element.
• On the other hand, a group is called centerless if its center only contains the identity element. The identity element is like the number zero in addition, or the number one in multiplication – it doesn't change anything when combined with other elements.

The elements that belong to the center are sometimes called central elements.

What Makes the Center a Subgroup?

The center of a group, Z(G), is always a subgroup of the main group G. This means it follows three important rules:

• It includes the identity: The identity element (let's call it e) is always in the center. This is because combining e with any element g always gives g, no matter the order (eg = g = ge).
• It's closed: If you pick any two elements from the center, say x and y, and combine them (xy), the result (xy) will also be in the center. This means it will commute with every other element in the main group.
• It includes inverses: If an element x is in the center, then its inverse element (x⁻¹) is also in the center. The inverse of x is the element that, when combined with x, gives the identity. If x commutes with everything, its inverse does too.

Because all elements in the center commute with each other, the center itself is always an abelian group. It is also a normal subgroup, which means it's "well-behaved" when you look at how it interacts with the rest of the group.

Examples of Centers

Let's look at some examples to understand the center better:

• Abelian Group: For any abelian group, where all elements already commute with each other, the center is the entire group itself.
• Dihedral Group: The dihedral group, D_n, represents the symmetries of a regular polygon with n sides (like rotations and flips).
  • If n is an odd number (like a triangle, D₃), the center is just the identity element.
  • If n is an even number (like a square, D₄), the center includes the identity element and the 180° rotation of the polygon.
• Quaternion Group: The quaternion group, Q₈ = {1, −1, i, −i, j, −j, k, −k}, has a center of {1, −1}. These are the only elements that commute with all others.
• Symmetric Group: The symmetric group, S_n, describes all possible ways to arrange n different items. For n greater than or equal to 3, the center of the symmetric group is just the identity element.
• Megaminx Group: The group of moves for a megaminx puzzle (a 12-sided Rubik's Cube-like puzzle) has a center that is a cyclic group of order 2. This means there are two elements in its center, one of which is the identity.

See also

• Group (mathematics)
• Subgroup (mathematics)
• Normal subgroup