Abstract
The group 27: G2(2), a maximal subgroup of the automorphism group of the Fischer group Fi22, is a semi-direct product of an elementary abelian group of order 27 and the exceptional group G2(2). This study aims to comprehensively determine and analyze the conjugacy classes of 27: G2(2) using computational methods. The conjugacy classes of a group provide valuable insights into its structure, symmetry, and representation theory, and have potential applications in various areas of mathematics, such as coding theory, cryptography, and mathematical physics. Using the computer algebra systems GAP and MAGMA, we computed the complete set of conjugacy classes of 27: G2(2) and analyzed their structure, sizes, and power maps. The results reveal that 27: G2(2) has 60 conjugacy classes, with sizes ranging from 1 to 110,592, and a non-uniform distribution of classes by element order. Comparison with the conjugacy classes of G2(2) shows that the extension by 27 leads to a splitting of conjugacy classes, with varying splitting patterns for different element orders. This study contributes significantly to the understanding of the structure of 27: G2(2) and lays the foundation for further exploration of its properties and applications. The conjugacy class information obtained in this research can be used in the study of representation theory, coding theory, cryptography, and mathematical physics. Our findings also provide insights into the relationship between 27: G2(2), its constituent group G2(2), and the larger context of the Fischer group Fi22 and its automorphism group.
Keywords
Conjugacy Classes, the Group 27: G2(2)
Citations
IRE Journals:
Rose Khadioli , Lucy Chikamai , Shem Aywa
"Conjugacy Classes of the Group 27: G2(2)" Iconic Research And Engineering Journals Volume 8 Issue 2 2024 Page 94-98
IEEE:
Rose Khadioli , Lucy Chikamai , Shem Aywa
"Conjugacy Classes of the Group 27: G2(2)" Iconic Research And Engineering Journals, 8(2)
About IRE Journals
IRE Journals is an open access online journals established with an aim to publish high quality of research work in various diciplines. We have more than 6 years in publications. Over 2500+ papers already published.
Important Links
For Authors
Volume 7:
Issue 1, Issue 2, Issue 3, Issue 4, Issue 5, Issue 6, Issue 7, Issue 8, Issue 9, Issue 10, Issue 11, Issue 12Volume 6:
Issue 1, Issue 2, Issue 3, Issue 4, Issue 5, Issue 6, Issue 7, Issue 8, Issue 9, Issue 10, Issue 11, Issue 12Volume 5:
Issue 1, Issue 2, Issue 3, Issue 4, Issue 5, Issue 6, Issue 7, Issue 8, Issue 9, Issue 10, Issue 11, Issue 12Volume 4:
Issue 1, Issue 2, Issue 3, Issue 4, Issue 5, Issue 6, Issue 7, Issue 8, Issue 9, Issue 10, Issue 11, Issue 12Volume 3:
Issue 1, Issue 2, Issue 3, Issue 4, Issue 5, Issue 6, Issue 7, Issue 8, Issue 9, Issue 10, Issue 11, Issue 12