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Complete factorizations of finite groups

Mikhail Kabenyuk

Abstract

Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the following theorem: Let $G$ be a finite nilpotent group. If $|G|=m_1\ldots m_k$ where $m_1,\ldots,m_k$ are integers greater $1$ and $k\geq3$, then there exist subsets $A_1,\ldots,A_k$ of $G$ which form a complete factorization of group $G$ and $|A_i|=m_i$ for all $i=1,2,\ldots,k$. In addition, we give several examples of building complete factorization for some groups and formulate one open question.

Complete factorizations of finite groups

Abstract

Let be a group. The subsets of form a complete factorization of group if if they are pairwise disjoint and each element is uniquely represented as , with . We prove the following theorem: Let be a finite nilpotent group. If where are integers greater and , then there exist subsets of which form a complete factorization of group and for all . In addition, we give several examples of building complete factorization for some groups and formulate one open question.
Paper Structure (4 sections, 3 theorems, 38 equations)

This paper contains 4 sections, 3 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem
  3. Examples
  4. Question

Key Result

Theorem 1

Let $G$ be a finite nilpotent group. If $|G|=m_1\ldots m_k$ where $m_1,\ldots,m_k$ are integers greater $1$ and $k\geq3$, then there exist subsets $A_1,\ldots,A_k$ of $G$ which form a complete factorization of group $G$ and $|A_i|=m_i$ for all $i=1,2,\ldots,k$.

Theorems & Definitions (9)

  • Theorem : On the complete factorization of finite nilpotent groups
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Remark 5
  • Example 6
  • Corollary 8
  • Lemma 9