Question

Let \(G\) be a group of order \(36 .\) If \(G\) has an element \(a \in G\) such that \(a^{12} \neq e\) and \(a^{18} \neq e,\) show that \(G\) is cyclic.

Short answer

Since the element \(a\) has order 36, \(G\) is cyclic.

Step-by-step solution

Understand Group Order

The group order is given as 36, which means the number of elements in the group is 36. Since group order can determine possible orders of group elements, this is important for our analysis of whether the group is cyclic.

Use Lagrange's Theorem

Lagrange's Theorem states that the order of any subgroup of a group divides the order of the group. Hence, any element's order, which generates a subgroup, must divide 36. Possible orders for elements are divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Interpret Conditions for Element Order

We have an element \(a\) such that \(a^{12} eq e\) and \(a^{18} eq e\), meaning the order of \(a\) cannot be 1, 2, 3, 4, 6, 12, or 18 as these would imply \(a^{12} = e\) or \(a^{18} = e\). Thus, the order of \(a\) must be the remaining possible value, 36.

Conclude Group Cyclicity

An element with the same order as the group itself (order 36) is maximal and generates the whole group. Thus, if \(a\) has an order of 36, it follows that \(G\) is cyclic and isomorphic to the cyclic group \( \mathbb{Z}_{36} \).

Key concepts

Understanding Group Theory

Group theory is a fascinating branch of algebra that studies the algebraic structures known as groups. At its core, a group is a set equipped with an operation that combines any two of its elements to form a third element, while adhering to four main properties: closure, associativity, identity, and invertibility.

• Closure: Performing the group operation on any two elements in the group results in another element within the group.
• Associativity: The group operation is associative, meaning for any elements \(a, b,\) and \(c\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity: There exists an element called the identity element, denoted as \(e\), which leaves other elements unchanged under the group operation (\(a \cdot e = e \cdot a = a\)).
• Invertibility: For every element \(a\) in the group, there is an inverse element \(b\) such that \(a \cdot b = b \cdot a = e\).

These properties form the cornerstone of group theory and are fundamental in understanding more complex structures and behaviors in abstract algebra.

Lagrange's Theorem and Its Implications

Lagrange's Theorem is a pivotal result in group theory that provides insight into the structure of a group. The theorem states that for any finite group \(G\), the order (or size) of every subgroup \(H\) of \(G\) divides the order of \(G\). This has immediate implications:

• It tells us that possible subgroup orders are restricted to the divisors of the group's order. For a group \(G\) of order 36, any subgroup must have an order that is a divisor of 36.
• If an element \(a\) of \(G\) has a certain order, the subgroup generated by \(a\) must also have an order that divides 36.

Understanding this theorem helps us determine that if an element in the group can generate a subgroup that is as large as the entire group, the group is cyclic.

Decoding Element Order

In group theory, the order of an element \(a\) in a group is the smallest positive integer \(n\) such that \(a^n = e\), where \(e\) is the group's identity element. This order tells us how many times the element must be combined with itself to return to the identity.

• An element that matches the group's order means this element, when used repeatedly in the group operation, cycles through all elements of the group.
• In our problem, the condition \(a^{12} eq e\) and \(a^{18} eq e\) implies the order of \(a\) is not 1, 2, 3, 4, 6, 12, or 18. Therefore, it must be 36.

Understanding element order is crucial as it helps identify if an element can generate the entire group, leading to the conclusion about the group's cyclic nature.

Concept of Isomorphism in Group Theory

Isomorphism in group theory is a fascinating concept that describes a relationship between two groups displaying the same structure. Two groups \(G\) and \(H\) are isomorphic (denoted as \(G \cong H\)) if there exists a bijective map (or function) between them that preserves the group operation.

• Isomorphisms reveal that two groups, while they may look different at first glance, have the same structural characteristics in terms of group operations.
• In the solution, when we show that a group \(G\) of order 36 is cyclic, it implies it is structurally identical to \(\mathbb{Z}_{36}\), the cyclic group of integers modulo 36.
• This means that solving problems in \(G\) can be simplified by considering the simpler structure of \(\mathbb{Z}_{36}\).

Understanding isomorphism helps mathematicians comprehend the essence of groups in greater depth, facilitating problem-solving and the discovery of new mathematical insights.