Question

Show that any cyclic group is Abelian. Hint: Does a matrix commute with itself?

Short answer

Any cyclic group is Abelian because the commutative property of addition in the exponents ensures that a * b = b * a for all elements.

Step-by-step solution

Define a Cyclic Group

A cyclic group is a group that can be generated by a single element. This means every element of the group can be written as powers (or multiples, in the case of additive notation) of this generator. Let the cyclic group be denoted by G and the generator by g.

Write Elements of the Cyclic Group

In a cyclic group G generated by g, any element in G can be written as g^n for some integer n. So, consider two arbitrary elements a and b in G such that a = g^m and b = g^n for some integers m and n.

Calculate the Product of Two Elements

Calculate the product of the two elements a and b. Using the properties of exponents, we have: a * b = g^m * g^n = g^(m+n).

Calculate the Product in Reverse Order

Now, calculate the product of the two elements in reverse order: b * a = g^n * g^m = g^(n+m).

Compare the Two Products

Notice that g^(m+n) = g^(n+m) due to the commutative property of addition in the exponents. Therefore, a * b = b * a.

Conclude the Group is Abelian

Since the product of any two elements a and b in the group is the same regardless of the order (a * b = b * a), the group is Abelian. Thus, any cyclic group is Abelian.

Key concepts

Abelian Group

An Abelian group is a type of group in mathematics where the operation of combining two elements is commutative, which means the order in which you combine them doesn't matter. If you have two elements, say, $a$ and $b$ in a group, then the group is Abelian if $a\ast b=b\ast a$. Abelian groups are named after the mathematician Niels Henrik Abel. These groups are fundamental in various areas of mathematics and are also known as commutative groups because of this property.

In the given exercise, you proved that cyclic groups are Abelian by showing that the generator of the group follows this rule. This is a key aspect of group theory as it simplifies many problems when you know the group is commutative.

Commutative Property

The commutative property is a fundamental principle in mathematics that applies to addition and multiplication. It states that changing the order of the numbers involved does not change the resulting value. For addition, it's written as $a+b=b+a$, and for multiplication, it's $a\times b=b\times a$.

In the context of group theory, this property extends to the operation defined by the group. For example, if the group operation is denoted by *, then for any elements $a$ and $b$ in the group, the group is commutative if $a\ast b=b\ast a$.

• In the provided exercise, you see this property in action when comparing $g^{m+n}$ and $g^{n+m}$. Because addition is commutative, $m+n$ is the same as $n+m$, making the group operation commutative.

This property is crucial for understanding why cyclic groups are Abelian.

Group Theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set of elements combined with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility.

These are explained as follows:

• Closure: If $a$ and $b$ are in the group, then $a\ast b$ is also in the group.
• Associativity: For any elements $a$, $b$, and $c$, $(a\ast b)\ast c=a\ast (b\ast c)$.
• Identity: There is an element $e$ in the group such that for any element $a$, $a\ast e=e\ast a=a$.
• Invertibility: For each element $a$, there is an element $b$ such that $a\ast b=b\ast a=e$, where $e$ is the identity element.

The exercise demonstrates that cyclic groups meet these criteria and are specifically Abelian because of the commutative property.

Understanding these concepts within group theory helps in grasping more advanced topics in mathematics and theoretical computer science.