RANDOM GENERATION IN FINITE GROUPS Mariapia Moscatiello University - PowerPoint PPT Presentation

RANDOM GENERATION IN FINITE GROUPS Mariapia Moscatiello University of Padova Young Researchers Algebra Conference 2019 Napoli 16th-18th September 2019 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION finite group ⋅ G M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION finite group ⋅ G ⋅ ( x k ) k ∈ℕ sequence of uniformly distributed - valued random variables G M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION finite group ⋅ G ⋅ ( x k ) k ∈ℕ sequence of uniformly distributed - valued random variables G Define a random variable: τ G = min { k ≥ 1 | ⟨ x 1 , …, x k ⟩ = G } M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION finite group ⋅ G ⋅ ( x k ) k ∈ℕ sequence of uniformly distributed - valued random variables G Define a random variable: τ G = min { k ≥ 1 | ⟨ x 1 , …, x k ⟩ = G } e ( G ) = ∑ kP ( τ G = k ) Expectation of τ G k ≥ 0 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION finite group ⋅ G ⋅ ( x k ) k ∈ℕ sequence of uniformly distributed - valued random variables G Define a random variable: τ G = min { k ≥ 1 | ⟨ x 1 , …, x k ⟩ = G } The expected number of e ( G ) = ∑ elements of G which have to be kP ( τ G = k ) drawn at random, with k ≥ 0 replacement, before a set of generators is found M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION we get τ G > k ⟺ ⟨ x 1 , …, x k ⟩ ≠ G Since P ( τ G > k ) = 1 − P G ( k ), the probability that k P G ( k ) = | {( g 1 , …, g k ) : ⟨ g 1 , …, g k ⟩ = G } | with randomly chosen | G | k elements generate G M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

INTRODUCTION we get τ G > k ⟺ ⟨ x 1 , …, x k ⟩ ≠ G Since P ( τ G > k ) = 1 − P G ( k ), the probability that k P G ( k ) = | {( g 1 , …, g k ) : ⟨ g 1 , …, g k ⟩ = G } | with randomly chosen | G | k elements generate G = ∑ kP ( τ G = k ) = ∑ P ( τ G = m ) ∑ e ( G ) k ≥ 1 k ≥ 1 m ≥ k = ∑ P ( τ G ≥ k ) = ∑ P ( τ G > k ) k ≥ 1 k ≥ 0 ( 1 − P G ( k ) ) ∑ = k ≥ 0 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE is a cyclic group of prime order p , then If G = C p is a geometric random τ G p variable of parameter p − 1 , so e ( C p ) = p − 1 p M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE Let G = D 2 p be the dihedral group of order 2p for an odd prime p G = ⟨ x 1 , …, x n ⟩ ⟺ ∃ 1 ≤ i < j ≤ n : x i ≠ 1 and x j ∉ ⟨ x i ⟩ M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE Let G = D 2 p be the dihedral group of order 2p for an odd prime p G = ⟨ x 1 , …, x n ⟩ ⟺ ∃ 1 ≤ i < j ≤ n : x i ≠ 1 and x j ∉ ⟨ x i ⟩ The number of trials needed to obtain x in G is a geometric random variable ≠ 1 2 p 2 p − 1 with parameter and expectation E 0 = 2 p − 1 ; 2 p M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE Let G = D 2 p be the dihedral group of order 2p for an odd prime p G = ⟨ x 1 , …, x n ⟩ ⟺ ∃ 1 ≤ i < j ≤ n : x i ≠ 1 and x j ∉ ⟨ x i ⟩ The number of trials needed to obtain x in G is a geometric random variable ≠ 1 2 p 2 p − 1 with parameter and expectation E 0 = 2 p − 1 ; 2 p p x has order the number of trials needed to With probability p 1 = 2 p − 1, 2 : 2 p find is a geometric with expectation y ∉ ⟨ x ⟩ E 1 = 2 p − 2 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE Let G = D 2 p be the dihedral group of order 2p for an odd prime p G = ⟨ x 1 , …, x n ⟩ ⟺ ∃ 1 ≤ i < j ≤ n : x i ≠ 1 and x j ∉ ⟨ x i ⟩ The number of trials needed to obtain x in G is a geometric random variable ≠ 1 2 p 2 p − 1 with parameter and expectation E 0 = 2 p − 1 ; 2 p p x has order the number of trials needed to With probability p 1 = 2 p − 1, 2 : 2 p find is a geometric with expectation y ∉ ⟨ x ⟩ E 1 = 2 p − 2 p 2 = p − 1 x has order the number of trials needed to p : With probability 2 p − 1, 2 p is a geometric with expectation find y ∉ ⟨ x ⟩ E 2 = 2 p − p M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE Let G = D 2 p be the dihedral group of order 2p for an odd prime p G = ⟨ x 1 , …, x n ⟩ ⟺ ∃ 1 ≤ i < j ≤ n : x i ≠ 1 and x j ∉ ⟨ x i ⟩ The number of trials needed to obtain x in G is a geometric random variable ≠ 1 2 p 2 p − 1 with parameter and expectation E 0 = 2 p − 1 ; 2 p p x has order the number of trials needed to With probability p 1 = 2 p − 1, 2 : 2 p find is a geometric with expectation y ∉ ⟨ x ⟩ E 1 = 2 p − 2 p 2 = p − 1 x has order the number of trials needed to p : With probability 2 p − 1, 2 p is a geometric with expectation find y ∉ ⟨ x ⟩ E 2 = 2 p − p 2 p 2 e ( D 2 p ) = E 0 + p 1 E 1 + p 2 E 2 = 2 + (2 p − 1)(2 p − 2) M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE Let G = D 2 p be the dihedral group of order 2p for an odd prime p G = ⟨ x 1 , …, x n ⟩ ⟺ ∃ 1 ≤ i < j ≤ n : x i ≠ 1 and x j ∉ ⟨ x i ⟩ The number of trials needed to obtain x in G is a geometric random variable ≠ 1 2 p 2 p − 1 with parameter and expectation E 0 = 2 p − 1 ; 2 p p x has order the number of trials needed to With probability p 1 = 2 p − 1, 2 : 2 p find is a geometric with expectation y ∉ ⟨ x ⟩ E 1 = 2 p − 2 p 2 = p − 1 x has order the number of trials needed to p : With probability 2 p − 1, 2 p is a geometric with expectation find y ∉ ⟨ x ⟩ E 2 = 2 p − p 2 p 2 e ( D 2 p ) = E 0 + p 1 E 1 + p 2 E 2 = 2 + (2 p − 1)(2 p − 2) In particular e ( Sym (3)) = 29 10 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

MÖBIUS FUNCTION The Möbius function on the subgroup lattice of G is defined as: μ G μ G ( G ) = 1 μ G ( H ) = − ∑ ∀ H < G μ G ( K ), H < K M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

MÖBIUS FUNCTION The Möbius function on the subgroup lattice of G is defined as: μ G μ G ( G ) = 1 μ G ( H ) = − ∑ ∀ H < G μ G ( K ), H < K Theorem (P . Hall) P G ( t ) = ∑ μ G ( H ) | G : H | t H ≤ G M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

MÖBIUS FUNCTION The Möbius function on the subgroup lattice of G is defined as: μ G μ G ( G ) = 1 μ G ( H ) = − ∑ ∀ H < G μ G ( K ), H < K Theorem (P . Hall) P G ( t ) = ∑ μ G ( H ) | G : H | t H ≤ G Theorem (A. Lucchini) e ( G ) = − ∑ μ G ( H ) | G | | G | − | H | H < G M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE μ G ( G ) = 1 μ G ( H ) = − ∑ μ G ( K ), ∀ H < G Sym (3) H < K ⟨ (123) ⟩ ⟨ (12) ⟩ ⟨ (13) ⟩ ⟨ (23) ⟩ {1} M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE μ G ( G ) = 1 μ G ( H ) = − ∑ μ G ( K ), ∀ H < G Sym (3) 1 H < K ⟨ (123) ⟩ ⟨ (12) ⟩ ⟨ (13) ⟩ ⟨ (23) ⟩ {1} M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE μ G ( G ) = 1 μ G ( H ) = − ∑ μ G ( K ), ∀ H < G Sym (3) 1 H < K ⟨ (123) ⟩ − 1 ⟨ (12) ⟩ ⟨ (13) ⟩ ⟨ (23) ⟩ − 1 − 1 − 1 {1} M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE μ G ( G ) = 1 μ G ( H ) = − ∑ μ G ( K ), ∀ H < G Sym (3) 1 H < K ⟨ (123) ⟩ − 1 ⟨ (12) ⟩ ⟨ (13) ⟩ ⟨ (23) ⟩ − 1 − 1 − 1 {1} 3 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

EXAMPLE μ G ( G ) = 1 μ G ( H ) = − ∑ μ G ( K ), ∀ H < G Sym (3) 1 H < K ⟨ (123) ⟩ − 1 ⟨ (12) ⟩ ⟨ (13) ⟩ ⟨ (23) ⟩ − 1 − 1 − 1 {1} 3 μ Sym (3) ( H ) | Sym (3) | ∑ e ( Sym (3)) = − | Sym (3) | − | H | H < Sym (3) 6 − 3 = 29 = − 3 ⋅ 6 6 6 6 − 1 + 3 ⋅ 6 − 2 + 10 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

WHAT IS KNOWN Dixon proved that e ( Sym ( n )) → 2.5 and e ( Alt ( n )) → 2, n → ∞ M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

WHAT IS KNOWN Dixon proved that e ( Sym ( n )) → 2.5 and e ( Alt ( n )) → 2, n → ∞ More generally for a finite, non abelian, simple group S , famous results of Dixon, Kantor-Lubotzky and Liebeck-Shalev establish that P S (2) → 1, | S | → ∞ From this one can deduce that e ( S ) → 2, | S | → ∞ M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS

WHAT IS KNOWN Dixon proved that e ( Sym ( n )) → 2.5 and e ( Alt ( n )) → 2, n → ∞ More generally for a finite, non abelian, simple group S , famous results of Dixon, Kantor-Lubotzky and Liebeck-Shalev establish that P S (2) → 1, | S | → ∞ From this one can deduce that e ( S ) → 2, | S | → ∞ Lucchini proved that for a non abelian, simple group S , e ( S ) ≤ e ( Alt (6)) ∼ 2.494 and that for n ≥ 5, 2.5 ≤ e ( Sym ( n )) ≤ e ( Sym (6)) ∼ 2.8816 M ARIAPIA M OSCATIELLO RANDOM GENERATION IN FINITE GROUPS