[Paper Review] Recent Progress in the Symmetric Generation of Groups

Ben Fairbairn|arXiv (Cornell University)|Apr 13, 2010

Finite Group Theory Research20 references3 citations

TL;DR

This paper surveys recent advances in symmetric generation of finite groups, focusing on sporadic simple groups and reflection groups using progenitors—semidirect products of free products of groups with control groups acting via automorphisms. It presents compact, symmetric presentations for many sporadic groups and extends the method to monomial progenitors using monomial group representations, offering elegant constructions and new insights into group structures, especially for groups like the Mathieu groups and others previously lacking symmetric presentations.

ABSTRACT

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and practically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmetric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a survey article originally accepted for inclusion in the proceedings of the 2009 `Groups St Andrews' conference. Since the article was accepted the author has become aware of other recent work in the subject that we incorporate to provide an updated version here (the most notable addition being the contents of Section 3.4.)

Motivation & Objective

• To update and expand upon earlier surveys of symmetric generation, particularly focusing on recent progress in constructing sporadic simple groups.
• To extend symmetric generation techniques beyond involutions to include non-involution symmetric generators, especially using monomial representations.
• To provide a comprehensive overview of symmetric presentations for reflection groups and finite simple groups, including new results on Coxeter groups and groups of Lie type.
• To unify and systematize the theory of symmetric generation using progenitors and monomial actions, especially through the use of monomial representations of control groups.
• To identify open problems and conjectures, particularly for the largest sporadic groups like the Monster and Baby Monster, which still lack known symmetric presentations.

Proposed method

• The paper uses progenitors of the form $ H^{\star n} \colon N $, where $ H $ is a group (often of order 2), and $ N $ acts on $ n $ copies of $ H $ via automorphisms.
• It employs monomial representations of control groups $ N $ to define actions on free products $ H^{\star n} $, leading to monomial progenitors $ H^{\star n} \colon_m N $.
• Relations are imposed in the form $ \pi w = \text{id} $, where $ \pi \in N $ and $ w $ is a word in symmetric generators, to factor the progenitor into a target group $ G $.
• Coset enumeration techniques are used to determine finiteness and structure of the resulting groups, particularly by analyzing double cosets $ NgN $.
• The method leverages known group actions and combinatorial designs—such as the $ \mathcal{S}(5,8,24) $ Steiner system for $ M_{24} $—to construct symmetric presentations.
• It applies results from monomial representation theory, including classifications of irreducible monomial representations of symmetric groups, alternating groups, and sporadic simple groups.

Experimental results

Research questions

• RQ1 How can symmetric generation be extended beyond involutory generators to include non-involution symmetric generators?
• RQ2 Which finite simple groups, particularly sporadic groups, admit symmetric presentations using monomial progenitors?
• RQ3 What role do monomial representations of control groups play in constructing symmetric presentations of reflection groups and other finite groups?
• RQ4 Can symmetric generation techniques be systematically applied to groups of Lie type, especially $ L_2(q) $, using elementary methods?
• RQ5 Why do some large sporadic groups like the Monster and Baby Monster still lack known symmetric presentations despite progress in other cases?

Key findings

• The paper presents updated symmetric presentations for several sporadic simple groups, including new constructions for the Rudvalis group, based on recent work not included in earlier surveys.
• Monomial progenitors defined via monomial representations of control groups yield symmetric presentations for a wide class of reflection groups and finite simple groups.
• The Mathieu group $ M_{24} $ is shown to act 5-transitively on 24 points and is deeply connected to the $ \mathcal{S}(5,8,24) $ Steiner system, which underpins symmetric generation constructions.
• The classification of irreducible monomial representations of $ L_2(q) $ and their covers enables symmetric presentations for these groups using elementary techniques.
• Despite progress, the Thompson group $ Th $, the Baby Monster $ \mathbb{B} $, and the Monster $ \mathbb{M} $ remain without known symmetric presentations, though conjectures exist.
• The method successfully constructs elegant, compact presentations for many sporadic groups, including the Conway groups $ Co_2 $ and $ Co_3 $, and the Lyons group $ Ly $, via symmetric generation.