id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-5889	Ejegwa, Paul Augustine ; Kausar, Nasreen; Musa Adeku Ibrahim; Cagin, Tonguc	Jordan-Hölder Theorem for Multigroups	2025	13	.pdf	application/pdf	5341	360	80	Math, 18 (2) (2025), 5889 6 of 13 Table 2: Submultigroups of D based on Case (iii) Submultigroups and their structures D0 = {1}, D1 = {1, d2}, D2 = D̂1 = {1, d, d2, d3}, D̂2 = {1, 1, d, d2, d3}, D̂3 = {1, 1, d, d, d2, d2, d3, d3}, D̂4 = {1, 1, 1, d, d2, d3}, D̂5 = {1, 1, 1, d, d, d2, d2, d3, d3}, D̂6 = {1, 1, 1, d, d, d2, d2, d2, d3, d3}, D̂7 = {1, 1, 1, 1, d, d2, d3}, D̂8 = {1, 1, 1, 1, d, d, d2, d2, d3, d3}, D̂9 = D = {1, 1, 1, 1, d, d, d2, d2, d2, d3, d3}, Ḋ1 = {1, 1}, Ḋ2 = {1, 1, 1}, Ḋ3 = {1, 1, 1, 1}, Ḋ4 = {1, 1, d2}, Ḋ5 = {1, 1, d2, d2}, Ḋ6 = {1, 1, 1, d2}, Ḋ7 = {1, 1, 1, d2, d2}, Ḋ8 = {1, 1, 1, d2, d2, d2}, Ḋ9 = {1, 1, 1, 1, d2}, Ḋ10 Math, 18 (2) (2025), 5889 7 of 13 Table 4: Submultigroups of E based on Case (iii) Submultigroups and their structures E0 = {ρ0}, Ê1 = E1 = {ρ0, ρ1, ρ2}, Ê2 = {ρ0, ρ0, ρ1, ρ2}, Ê3 = {ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}, Ê4 = {ρ0, ρ0, ρ0, ρ1, ρ2}, Ê5 = E = {ρ0, ρ0, ρ0, ρ1, ρ1, ρ2, ρ2}, Ė1 = {ρ0, ρ0}, Ė2 = {ρ0, ρ0, ρ0} Here, E0, Ê1, Ė1, Ė2 and Ê5 are trivial, and Ê2–Ê4 are proper non-trivial normal submultigroups of E without a maximal non-trivial normal submultigroup of E since either	cache/ejpam-5889.pdf	txt/ejpam-5889.txt