On the covering number of small alternating groups

Kappe, Luise-Charlotte; Redden, Joanne L.

doi:10.1090/conm/511/10045

Cited by 11 publications

(16 citation statements)

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“…We follow the notation from Kappe and Redden [9] for the disjoint cycle decomposition of a nontrivial permutation. Let 1 2 , ,....., k n n n N with 1 2 1 ..... k n n n and 1 2 , ,....., k t t t N .…”

Section: Notationmentioning

confidence: 99%

“…This is the method used by Holmes [5] to find the minimal covering of some of sporadic groups. Followed by Kappe and Redden [9] to determine minimal covering of alternating groups, A n for n = 7, 8, 9 and 10. Here, we applied this method to find minimal covering of symmetric group of degree 9, S 9 .…”

Section: The Symmetric Group Smentioning

confidence: 99%

“…For the alternating group, he proved that if n ≠ 7, 9, then σ(A n ) ≥ 2 n-2 and with equality if and only if n is even but not divisible by 4. In 2010, Kappe and Redden [9] determined σ(A n ) for n = 7, 8, 9 and 10 by applying the method used in [5].…”

Section: Introductionmentioning

confidence: 99%

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Subgroup coverings of symmetric group of degree nine

Tarmizi¹,

Sulaiman²

2014

AIP Conference Proceedings

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Let G be a finite group. A covering of G is a set of proper subgroups, whose union is equal to G. The least number of covering is denoted by σ(G) and a covering of cardinality σ(G) is called a minimal covering. In this paper, we investigate the minimal covering of symmetric group of degree nine, S 9 . In finding the minimal covering of a group, we only need to consider the number of maximal subgroups of a group. We used Group Algorithm Programming (GAP) to find the conjugacy classes of maximal subgroups for S 9 . In order to determine the minimal covering of S 9 , it suffices to find a minimal covering of maximal cyclic subgroups by maximal subgroups of S 9 . We give a proof that 242 ≤ σ( S 9 ) ≤ 256.

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Section: Notationmentioning

confidence: 99%

Section: The Symmetric Group Smentioning

confidence: 99%

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Subgroup coverings of symmetric group of degree nine

Tarmizi¹,

Sulaiman²

2014

AIP Conference Proceedings

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“…It is a natural question to ask what are the exact covering numbers for alternating and symmetric groups for those values of n where Maróti only gives estimates. In [17] and [8] this was done for alternating groups in case of small values of n. As mentioned earlier, Cohn [6] already established σ(A 5 ) = 10. In [17] it is shown that σ(A 7 ) = 31 and σ(A 8 ) = 71.…”

Section: Introductionmentioning

confidence: 97%

On the covering number of small symmetric groups and some sporadic simple groups

Kappe

Nikolova-Popova²,

Swartz

2016

Groups Complexity Cryptology

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ABSTRACT. A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ(G). Determining σ(G) is an open problem for many non-solvable groups. For symmetric groups S n , Maróti determined σ(S n ) for odd n with the exception of n = 9 and gave estimates for n even. In this paper we determine σ(S n ) for n = 8, 9, 10 and 12. In addition we find the covering number for the Mathieu group M 12 and improve an estimate given by Holmes for the Janko group J 1 .

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“…Small values of n have been resolved elsewhere. Cohn [6] showed that σ(A 5 ) = 10; Kappe and Redden [12] showed that σ(A 7 ) = 31, σ(A 8 ) = 71, and 127 σ(A 9 ) 157; and recently Epstein, Magliveras, and Nikolova-Popova [7] showed that σ(A 9 ) = 157 and σ(A 11 ) = 2751.…”

Section: Introductionmentioning

confidence: 99%

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

Swartz

2016

Discrete Mathematics

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If a group G is the union of proper subgroups H 1 , . . . , H k , we say that the collection {H 1 , . . . H k } is a cover of G, and the size of a minimal cover (supposing one exists) is the covering number of G, denoted σ(G). Maróti showed that σ(S n ) = 2 n−1 for n odd and sufficiently large, and he also gave asymptotic bounds for n even. In this paper, we determine the exact value of σ(S n ) when n is divisible by 6.

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On the covering number of small alternating groups

Cited by 11 publications

References 0 publications

Subgroup coverings of symmetric group of degree nine

Subgroup coverings of symmetric group of degree nine

On the covering number of small symmetric groups and some sporadic simple groups

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

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