Understanding the Frobenius Group of Order 21: A Non-Abelian Example
The Frobenius group F21 of order 21 is a fascinating example of a non-abelian group, providing a rich ground for exploring group theory concepts. In this article, we will delve into the specific details of this group, including its generation, structure, and presentation, and provide visual and mathematical insights to aid in understanding.
Generation and Structure
The Frobenius group F21 is unique because it can be generated by two permutations: a (1,2,3,4,5,6,7) and b (1,2,4,3,6,5), where the group is formed through the semi-direct product of the cyclic group C7 and the cyclic group C3. Here, C7 is the cyclic group of order 7 and C3 is the cyclic group of order 3.
This group can be viewed as a semidirect product C7rtimes C3, where the automorphism of C7 is given by the rule a to a^2, and the automorphism has order 3. The action of C3 on C7 is thus defined via this automorphism. Each element of F21 can be expressed as b^m a^n, where 0 ≤ m ≤ 2 and 0 ≤ n ≤ 6 (these values are derived from the orders of the respective cyclic groups). A crucial property of F21 is that a and b do not commute. Specifically, we have the relation bab^(-1) a^2.
Cayley Graph Visualization
The Cayley graph of F21 offers a visual representation of the group's structure. The Cayley graph for F21 has a unique feature where the two ends of the graph are the same. This implies that the graph should be rolled up into a cylinder. When we further consider the similarity of the top and bottom ends, we can visualize the graph as being rolled up into a torus. Such visual representations are valuable for gaining a deeper understanding of the group's topology and structure.
Group Presentation
A concise presentation of the Frobenius group F21 is as follows:
F21 a, b : a^31, b^71, aba^(-1)b^2
From this presentation, we can deduce that b is a normal subgroup of C7 and that the order of the element a acting on b via conjugation is 3. Additionally, the fact that 2^3 8 is congruent to 1 mod 7 plays a crucial role in understanding the group's properties and its semi-direct product structure.
Conclusion
The Frobenius group F21 is an exemplary case of a non-abelian group that captures the essence of semi-direct products and automorphisms in group theory. Through its generation by permutations, its presentation, and its visual representation via the Cayley graph, we can explore the intricate structure and properties of this group, which is both a theoretical and applied subject in various fields such as number theory, algebra, and cryptography.