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There is a quite elementary proof that uses only the fact that a polynomial of degree $n$ has at most $n$ roots in a field. So, let $F$ be a field and $G$ be a finite subgroup of the multiplicative gr …

This question is follow up of this MO-post. First let us recall the necessary definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0 …

This question is motivated by (some available information on) this MO-problem on the largest possible degree of a polynomial on a finite group and this MO-problem on the degree of the constant polynom …

A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N=\{1,2,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. T …

A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form $f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of th …

Realizing the idea of @NickGill we shall confirm the lower bound for solvable groups with five exceptions of the groups $G$ isomorphic to the groups $C_3,C_5,C_3\times C_3, D_{10}$ and $(C_3\times C_ …

For finite solvable groups $G$ we have the following lower bound for the number $f_1(G)$. Theorem. Let $G$ be a finite solvable group of cardinality $|G|=\prod_{k=1}p_k$ for some prime numbers $p_1,\d …

$\DeclareMathOperator\SmallGroup{SmallGroup}$Definition. A subset $A$ of a group $G$ is called product-1-free if for any sequence of pairwise distinct elements $a_1,\dots,a_n$ of $A$ the product $a_1\ …

GAP shows that the groups SmallGroup(27,3), SmallGroup(27,4), SmallGroup(36,11), SmallGroup(39,1) SmallGroup(48,3) do contain many 5-element decomposable sets, which are not product-one. So, the lower …

This is not an answer, but too long for a comment. Below I write down some conditions (on a group or a decomposable set) guaranteeing that a decomposable set in a group is product-one. Proposition 1. …

A nonempty subset $D$ of a group $G$ is called $\bullet$ decomposable if $D\subseteq DD$, that is every element $x\in D$ is can be written as the product $x=yz$ of some elements $y,z\in D$; $\bullet$ …

I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks. Le …

The following theorem gives a partial answer to Problem 2 on the structure of almost squared groups. Let us recall that the Abelianization of a group $G$ is the quotient group $G/[G,G]$ of $G$ by its …

The following theorem yields a partial answer to Problem 2. A subset $C$ of a group $G$ is called unfree if $xy=yx$ or $x^2=y^2$ for any elements $x,y\in C$. For a group $G$ let $ucov(G)$ be the small …

Definition. A finite group $G$ is called squared (resp. almost squared) if there exists a subset $A\subseteq G$ such that $G=\{ab:a,b\in A\}$ and $|G|=|A|^2$ (resp. $|G|=|A|^2-1$). Such a set $A$ wil …