Generation of groups

Minimal and special generating sets: generator numbers of almost simple and soluble groups, (2,3)-generation, generation by elements of prime-power order, spread, coprime generation, two-generated direct powers, Eulerian functions and probabilistic generation.

• (2,3)-generation of groups
• Coprime generation
• Eulerian polynomials of groups
• Generation by elements of prime-power order
• Generator number of almost simple groups
• Rank of a group generated by two subgroups
• Spread of a group
• Two-generated direct powers of simple groups

TheoremSubgroup generated by a symmetric set#✓fully proved

Let \(G\) be a group and let \(X \subseteq G\) satisfy \(X^{-1} = X\), where \(X^{-1} = \{x^{-1} : x \in X\}\). Then the subgroup of \(G\) generated by \(X\) equals \(\bigcup_{n \in \mathbb{N}} X^n\), where \(X^0 = \{1\}\) and \(X^{n+1} = X^n X = \{yx : y \in X^n,\ x \in X\}\).

theorem closure_eq_iUnion_pow (hX : X⁻¹ = X) :
    (Subgroup.closure X : Set G) = ⋃ n : ℕ, X ^ n

Sources

• 2310.10246 D01 (main.tex:146).

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