Finite Axiomatization of Finite Soluble Groups

Wilson, John S.

doi:10.1112/s0024610706023106

Cited by 17 publications

(14 citation statements)

References 13 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) is proved in the previous section and (2) is proved in [50]. Consider G = SU(3, q) with q = p e .…”

Section: Low Rank Groupsmentioning

confidence: 95%

See 1 more Smart Citation

Presentations of finite simple groups: profinite and cohomological approaches

Guralnick¹,

Kantor²,

Kassabov³

et al. 2007

Groups Geom. Dyn.

View full text Add to dashboard Cite

show abstract

“…(1) is proved in the previous section and (2) is proved in [50]. Consider G = SU(3, q) with q = p e .…”

Section: Low Rank Groupsmentioning

confidence: 95%

“…One may hope that 4 is the right upper bound for both types of presentations. Indeed, there is no known obstruction to the full covering group of a finite simple group having a presentation with 2 generators and 2 relations (see [50]). …”

Section: Profinite Versus Discrete Presentationsmentioning

confidence: 99%

Presentations of finite simple groups: profinite and cohomological approaches

Guralnick¹,

Kantor²,

Kassabov³

et al. 2007

Groups Geom. Dyn.

View full text Add to dashboard Cite

show abstract

“…As Thompson's classification of the minimal simple groups is a very useful tool to obtain solvability criteria in the class of finite groups (e.g., see [14] for a recent and interesting application of Thompson's result) we hope that these new results might be useful to obtain new solvability criteria.…”

Section: Zarrinmentioning

confidence: 99%

On element-centralizers in finite groups

Zarrin

2009

Arch. Math.

View full text Add to dashboard Cite

show abstract

“…However, definability of some group theoretic notions have been studied before: simplicity ( [3,5,2,9]), nilpotency [5,1], solvability [1,5,10], and the normal closure of a single element [5].…”

Section: Introductionmentioning

confidence: 99%

Descriptive Complexity of Finite Abelian Groups

Gomaa

2010

Int. J. Algebra Comput.

View full text Add to dashboard Cite

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G 1 and G 2 be a pair of nonisomorphic finite abelian groups. Then there exists a positive integer m that divides one of the two groups' orders such that the following holds: (1) there exists a first-order sentence ϕ that distinguishes G 1 and G 2 such that ϕ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if ϕ is a sentence that distinguishes G 1 and G 2 then ϕ must have quantifier depth Ω(log m). These results are applied to (1) get bounds on the first-order distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not first-order definable, and (3) give a different proof for the first-order undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by Koponen and Luosto in an unpublished paper). Descriptive Complexity of Finite Abelian Groups 1089are given in Sec. 3. In Sec. 4, EF games are applied to the groups Z p and Z q for prime numbers p and q to find bounds on the quantifier depth of a distinguishing first-order sentence. In Sec. 5, an extended version of EF games (using pebbles) is applied to the same groups to find bounds on the number of variables in a distinguishing first-order sentence. In Sec. 6, the game is applied to groups modulo any number. In Sec. 7 bounds are obtained for any finite abelian groups. In Sec. 8 we use the above bounds to get definability results on the following group-theoretic notions: cyclicity, simplicity, nilpotency, the closure of a single element, and dihedral groups. In Sec. 9 we state some of the open problems to look at. Ehrenfeucht-Fraïssé GamesAs described above EF games are used as a tool to get upper and/or lower bounds on logical expressibility. An EF -game [6, 4] is played over two structures of the same kind, for example two linear orderings. There are two players: the spoiler denoted by S and the duplicator denoted by D. The game has k rounds, for some non-negative integer k. Intuitively, the goal of S is to show that the two structures can be distinguished in at most k steps, whereas D wants to show that this cannot be done.Definition 2 (Partial isomorphism). Let A and B be two first-order structures with vocabulary τ . Assumeā = a 1 , . . . , a n ∈ A n andb = b 1 , . . . , b n ∈ B n . We say that there is a partial isomorphism fromā ontob if for every m, for every quantifierfree formula ϕ(x 1 , . . . , x m ) over τ , and for every multiset {i 1 , . . . , i m } ⊆ {1, . . . , n} the following holds:If A and B are groups, then partial isomorphism basically means that for every multiset {i 1 , i 2 , i 3 } ⊆ {1, . . . , n}, the following hold:We now des...

show abstract

Finite Axiomatization of Finite Soluble Groups

Cited by 17 publications

References 13 publications

Presentations of finite simple groups: profinite and cohomological approaches

Presentations of finite simple groups: profinite and cohomological approaches

On element-centralizers in finite groups

Descriptive Complexity of Finite Abelian Groups

Contact Info

Product

Resources

About