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Finite solvable tidy Groups whose orders are divisible by two primes

Nicolas F. Beike, Rachel Carleton, David G. Costanzo, Colin Heath, Mark L. Lewis, Kaiwen Lu, Jamie D. Pearce

Abstract

In this paper, we investigate finite solvable tidy groups. We classify the tidy $\{ p, q \}$-groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.

Finite solvable tidy Groups whose orders are divisible by two primes

Abstract

In this paper, we investigate finite solvable tidy groups. We classify the tidy -groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.
Paper Structure (5 sections, 23 theorems, 5 equations)

This paper contains 5 sections, 23 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Elements of Prime Power Order
  3. Some tidy groups
  4. Tidy $\{ p, q\}$-groups
  5. Solvable tidy groups

Key Result

Theorem 1

Let $G$ be a solvable, tidy group. Then $G$ has Fitting height at most $4$ and $G/F(G)$ has derived length at most $4$. If $|G|$ is odd, then $G$ has Fitting height at most $3$ and $G/F(G)$ is abelian or metabelian.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 30 more