Is any group of order 36 abelian?

Yes, there are groups of order 36 that are abelian.

Groups of a specific order can exhibit a wide range of properties, from being commutative (abelian) to highly non-commutative (non-abelian). For groups of order 36, mathematical classification reveals that both types exist.

Classification of Groups of Order 36

In total, there are 14 distinct isomorphism classes of groups with order 36. Among these 14 unique structures, a significant portion are indeed abelian, meaning their elements commute under the group operation.

• Abelian Groups: Specifically, 4 of the 14 groups of order 36 are abelian. These groups are characterized by the property that the order of operations does not affect the result (i.e., for any elements 'a' and 'b' in the group, a * b = b * a). Examples of abelian groups of order 36 include the cyclic group of order 36 (C₃₆) and direct products of smaller cyclic groups, such as C₁₈ × C₂ or C₆ × C₆.
• Non-Abelian Groups: The remaining 10 groups of order 36 are non-abelian. Interestingly, among these non-abelian groups, 4 are also classified as nilpotent. Nilpotent groups are a special class of non-abelian groups that possess a strong hierarchical structure, even though their elements do not generally commute.

Key Properties of Groups of Order 36

All groups of order 36, whether abelian or non-abelian, share certain fundamental characteristics:

• Solvability: Every group of order 36 is a solvable group. This property is a direct consequence of Burnside's pᵃqᵇ-theorem, which states that any finite group whose order is a product of at most two distinct prime powers is solvable. Since 36 = 2² * 3², it fits this criterion (p=2, q=3).
• Normal Sylow Subgroups: A notable feature of groups of order 36 is that each one must possess either a normal 2-Sylow subgroup or a normal 3-Sylow subgroup. This structural characteristic provides important insights into their composition and simplifies their study.

Summary of Group Types of Order 36

The table below summarizes the different types of groups found for order 36:

The existence of four distinct abelian groups among the 14 possible structures for groups of order 36 confirms that not all groups of this order are non-abelian.