Understanding Cyclic Groups and Abelian Groups

Let's analyze the given statements about groups, specifically focusing on the properties of a Cyclic Group and its relationship with an Abelian Group, as well as the concept of the Order of Group and the Order of Generator.

Analyzing Statement A: All cyclic groups are an abelian group.

A group $G$ is called a Cyclic Group if there exists an element $g$ in $G$ such that every element in $G$ can be written as a power of $g$. This element $g$ is called a generator of the group, and we denote the group as $G = \langle g \rangle$.

A group $G$ is called an Abelian Group (or commutative group) if for any two elements $a, b \in G$, the operation satisfies $ab = ba$.

Now, let's consider any two elements $x$ and $y$ in a Cyclic Group $G = \langle g \rangle$. Since $G$ is generated by $g$, both $x$ and $y$ must be powers of $g$. Let $x = g^a$ and $y = g^b$ for some integers $a$ and $b$.

To check if the group is abelian, we need to see if $xy = yx$.

Let's compute $xy$:

$xy = g^a g^b = g^{a+b}$ (using the property of exponents in group theory).

Now let's compute $yx$:

$yx = g^b g^a = g^{b+a}$ (using the property of exponents).

Since addition of integers is commutative, $a+b = b+a$. Therefore, $g^{a+b} = g^{b+a}$. This implies that $xy = yx$ for all $x, y$ in the Cyclic Group $G$.

Thus, any Cyclic Group satisfies the commutative property, which means all cyclic groups are indeed abelian groups. Statement A is true.

Analyzing Statement B: The order of the cyclic group is the same as the order of its generator.

The Order of Group $G$, denoted $|G|$, is the number of elements in the group.

The Order of Generator $g$, denoted $|g|$, is the smallest positive integer $n$ such that $g^n = e$, where $e$ is the identity element of the group. If no such positive integer exists, the order of the element is said to be infinite.

Consider a Cyclic Group $G = \langle g \rangle$ generated by an element $g$. The elements of $G$ are of the form $g^k$ for integers $k$. The distinct elements generated by $g$ are $g^0, g^1, g^2, \dots, g^{|g|-1}$, where $|g|$ is the order of the generator $g$.

There are exactly $|g|$ distinct elements in the set $\{g^0, g^1, g^2, \dots, g^{|g|-1}\}$. Any other power $g^k$ will be equal to one of these elements. For example, $g^{|g|} = g^0 = e$, $g^{|g|+1} = g^1$, and so on.

So, the number of elements in the Cyclic Group $G = \langle g \rangle$ is equal to the number of distinct powers of $g$, which is exactly $|g|$, the Order of Generator.

Therefore, the order of the Cyclic Group is the same as the order of its generator. Statement B is true.

Conclusion on Statements A and B

Based on our analysis using concepts from Group Theory and Abstract Algebra:

• Statement A: All cyclic groups are an abelian group. This is true.
• Statement B: The order of the cyclic group is the same as the order of its generator. This is true.

Both statement A and statement B are true.