On the covering number of symmetric groups having degree divisible by six

Swartz, Eric

doi:10.1016/j.disc.2016.05.004

Cited by 22 publications

(16 citation statements)

References 26 publications

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“…Proof. The proof is essentially identical to that of [15,Lemma 5.1] and [31,Lemma 3.1] with the word "group" changed to the word "ring," but it is included for the sake of completeness. Let D and Π be as in the statement of the lemma, and assume that c(M) 1 for all maximal subrings not in D; the proof with strict inequalities is analogous.…”

Section: 2mentioning

confidence: 94%

“…This conjecture was proven by Tomkinson [33], who also proved that there is no group with covering number equal to 7. The covering numbers have been determined for various families of simple and almost simple groups (see, for instance, [2,3,4,16,24,31]). As noted above, no group has covering number 7, and other natural numbers exist that are not the covering number of a group (the next smallest examples being 11 [10] and 19 [14]).…”

Section: Introductionmentioning

confidence: 99%

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The covering numbers of rings

Swartz¹,

Werner²

2021

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A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ(R) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ(R) < σ(R/I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σ u (R) to be the size of a minimal cover of R by subrings that contain 1 R (if such a cover exists), and R is σ u -elementary if σ u (R) < σ u (R/I) for every nonzero two-sided ideal of R. In this paper, we classify all σ-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σ u -elementary rings and prove σ u (R) = σ(R) for every σ u -elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R ′ such that σ(R) = σ u (R ′ ), which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if E (N ) := {m : m N, σ(R) = m for some ring R}, then we show that |E (N )| = Θ(N/ log(N )), which proves that almost all integers are not covering numbers of a ring.

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Section: 2mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

The covering numbers of rings

Swartz¹,

Werner²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The determination of σ (S n ) when n is even seems a more difficult task. In case n is divisible by 6, E. Swartz [16] managed to give a formula for σ (S n ). Apart from this case, the value of σ (S n ) for n even is only known for n ≤ 14: see [2,8,14].…”

Section: Introductionmentioning

confidence: 99%