In this work, we show that a very large class of matroid groups possesses the basis property. Moreover, we show that this class behaves like vector spaces in terms of basis. Applications include new proofs for the characterization of finite matroid groups. Moreover, we show that every group possesses BEP, also possesses the span property and in the deﬁnition of matroid group, the extension property can be replaced by BEP. The fact that BEP always correct in vector spaces, but the situation is diﬀerent in groups was showed. In the end, we show that each base and maximal independent subset are equivalent in any group with embedding property.
R. J. Wilson, "An Introduction to Matroid Theory," Amer. Math. Monthly, 5, pp. 500-525, 1973.
R. Scapellato, and L. Verardi, "Sur les ensembles generateurs minimaux d’un groupe ﬁni," Ann. Sci. Univ. B. Pascal, Clermont II, Ser. Mat., no. 26, pp. 51-60, 1990. (French).
R. Scapellato, and L. Verardi, "Groupes ﬁnis qui jouissent d’une propriete analogue au theoreme des bases de Burnside," Boll. Un. Mat. It.,vol. 7, no. 5-A, pp. 187-194, 1991. (French)
P. Jones, "Basis properties for inverse semigroups," J. Algebra, Vol. 50 (1978), 135-152.
A. Aljouiee, "Basis property conditions on some groups," Int. J. of Math. & Comp. Sci.,vol. 3, pp. 102-112, 2008.
A. Alkhalaf, "Finite groups with bassically property," Dokl. Akad. Nauk. BSSR, no. 11, pp. 47-56, 1989. (Russian).
J. McDougall-Bagnall, and M. Quick, "Groups with the basis property," J. Algebra,vol. 346, pp. 332–339, 2011.
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