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Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? Or under what condition this can be true? I am fine to assume $M$ to be a partition matroid(or even a uniform matroid) and also assume $G$ is cyclic of prime order. (A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser's theorem)

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? Or under what condition this can be true? I am fine to assume $M$ to be a partition matroid(or even a uniform matroid) and also assume $G$ is cyclic of prime order. (A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser's theorem)

Note: To make the statement plausible, adding the condition of $X+Y$ to be independent won't hurt.

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? (A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser's theorem)

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? Or under what condition this can be true? I am fine to assume $M$ to be a partition matroid(or even a uniform matroid) and also assume $G$ is cyclic of prime order. (A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser's theorem)

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of G, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H$=$X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? (A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser's theorem)

Assume that $G$ is a (finite) abelian group and $M$ is a matroid whose ground set is $G$. Let $X$ and $Y$ be subsets of $G$, and $H$ is the stabilizer of $X+Y$. That is $X+Y+H=X+Y$. We denote the rank function of $M$ by $r$. Then can we say that $r(X+Y)\geq$ $r(X)+r(Y)-r(H)$? (A similar result holds for the cardinality of sumsets of a given group, well-known as Kneser's theorem)