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Is there any finite group $G$ of order $n=m^2$ satisfying the following conditions? (a) There is no subgroup of order $m$; (b) There exist subsets $A$, $B$ such that $|A|=|B|=m$ and $G=AB$. Conside …

Let $G$ be a group with the property $G=G_e\dot{\cup} G_o$ with $G_oG_o\subseteq G_e\leq G$. ($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G_o^{-1}=\{x^{-1}: x\in G_o\}$) …

In our research, we need to know that whether every group $G$ of order $2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7=\frac{7!}{2}$ or $5040 = 2^4 \cdot 3^2 \cdot 5 \cdot 7=7!$ has a proper subgroup non-isomor …

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subgroup of $G$ with order $d$ or $n/d$? Of course this does not hold in full generality. -- In parti …

I want to write a GAP program for checking the following question. Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ …

Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subset $A$ of $G$ with $d$ elements and a subset $B$ such that $G=AB$ and $|AB|=|A||B|$ (equivalently …

Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$ It is clear that $1\leq c(A)\leq \frac{|G|}{|A|}$ …

During my researches, I've obtained a class of finite groups as follows. Let $\mathcal{C}$ be the class of all finite groups $G$ such that for every factorization $|G|=ab$ there exists a subgroup $H\n …