id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
cana-918	Anuradha Sabharwal	Association Schemes for Some Finite Group Rings II	2024	11	.pdf	application/pdf	5910	274	80	Table 1 Canonical forms of some general linear groups Group Generators Presentation Canonical form 𝐺𝐿(2, ℤ2) 𝐴, 𝐵 τ2 = σ3 = (τσ)3 = 1 𝐴𝑎𝐵𝑏: 0 ≤ 𝑎 ≤ 1, 0 ≤ 𝑏 ≤ 2 𝐺𝐿(2, ℤ4) 𝐴, 𝐵, 𝐶 𝐴2 = 𝐵2 = 𝐶4 = 𝑋3 = 𝑌4 = (𝐶𝐵𝐴)4 = 𝐼, 𝐶2𝐴 = 𝐴𝐶2, 𝐶2𝐵 = 𝐵𝐶2, 𝐵𝐶 = 𝐶−1𝐵 where 𝑋 = 𝐴𝐵 and 𝑌 = 𝑋𝐶𝑋 𝐴𝑎𝐶𝑏𝑌𝑐𝑋𝑑: 0 ≤ 𝑎 ≤ 1,0 ≤ 𝑏, 𝑐 ≤ 3, 0 ≤ 𝑑 ≤ 2 𝐺𝐿(2, ℤ6) 𝐴, 𝐵, 𝐶 𝐴2 = 𝐵2 = 𝐶4 = 𝑋12 = 𝑌6 = (𝐴𝐵)3 = (𝐵𝐶)2 = 𝐼, 𝐶𝐴 = 𝐴𝐶2, 𝐶2𝐵 = 𝐵𝐶2, (𝐴𝐶)12 = 𝐶2, (𝐶𝐴)2𝐵(𝐶𝐴)2 = (𝐴𝐶)2𝐵(𝐴𝐶)2 where 𝑋 = 𝐵(𝐶𝐴)2𝐵 and 𝑌 = 𝐶𝐴𝐵 𝐶𝑎𝑋𝑏𝑌𝑐𝐵𝑑: 0 ≤ 𝑎, 𝑑 ≤ 1, 0 ≤ 𝑏 ≤ 11, 0 ≤ 𝑐 ≤ 5 Theorem 4. Table 2 Canonical forms of alternating and symmetric groups of degree 4 and 5 Group Generators Presentation Canonical form 𝐴4 𝜏, 𝜎 𝐴2 = 𝐵3 = 𝐼, 𝐴𝐵 = 𝐵−1𝐴 𝜏𝑎𝜎𝜏𝑏𝜎𝑐: 0 ≤ 𝑎, 𝑏 ≤ 1, 0 ≤ 𝑐 ≤ 2 𝑆4 𝜏, 𝜎 𝜏2 = 𝜎3 = (𝜏𝜎)3 = 1 𝐴2𝜏𝑎𝜎𝑏𝜏𝜎𝑐: 0 ≤ 𝑎 ≤ 1, 0 ≤ 𝑏 ≤ 2, 0 ≤ 𝑐 ≤ 3 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31	cache/cana-918.pdf	txt/cana-918.txt