The simple group of order 604,800

Hall, Marshall; Wales, David B.

doi:10.1016/0021-8693(68)90014-8

Cited by 58 publications

(50 citation statements)

References 12 publications

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“…For the final case J7 3 (3) one must check a character table [16]. Here there are three representations of degree 7, two of them are conjugate and the third rational.…”

Section: David B Walesmentioning

confidence: 99%

Finite linear groups of degree seven. II

Wales¹

1970

Pacific J. Math.

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“…For the final case J7 3 (3) one must check a character table [16]. Here there are three representations of degree 7, two of them are conjugate and the third rational.…”

Section: David B Walesmentioning

confidence: 99%

Finite linear groups of degree seven. II

Wales¹

1970

Pacific J. Math.

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“…The permutation representation on 45 points is the permutation representation on the images of the orbit of length 2. The orbit lengths of the stabilizer of a point are 1,8,8,2,2,4,4,8,8. We must associate two orbits of length 4 with this point.…”

Section: δ(A) π γ(B) and γ(A) F] δ(B) Are Permutation Isomorphic As Amentioning

confidence: 99%

“…We next consider a rank three extension of £7 3 (3) on orbits of lengths 1, 36, and 63 described in [1]. As shown in [1] the permutation representation on 36 letters is unique.…”

Section: δ(A) π γ(B) and γ(A) F] δ(B) Are Permutation Isomorphic As Amentioning

confidence: 99%

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Uniqueness of the graph of a rank three group

Wales¹

1969

Pacific J. Math.

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“…Frobenius conjectured that Ln(G) forms a subgroup of G provided \Ln(G)\ = « (see [2]). Zemlin [25] has reduced the conjecture to the classification of finite simple groups which is now complete (see [8]). The author has verified the conjecture for the Fischer Griess monster Fx and the Fischer baby monster F2 in [24].…”

mentioning

confidence: 99%

A conjecture of Frobenius and the sporadic simple groups. II

Yamaki¹

1986

Math. Comp.

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Abstract. A conjecture of Frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups.Let G be a finite group and « be a positive integer dividing \G\. Let Ln(G) = {x e G\x" = 1). Then by a theorem of Frobenius [6] one knows that \Ln(G)\ = cnn for some integer cn. Frobenius conjectured that Ln(G) forms a subgroup of G provided \Ln(G)\ = « (see [2]). Zemlin [25] has reduced the conjecture to the classification of finite simple groups which is now complete (see [8]). The author has verified the conjecture for the Fischer Griess monster Fx and the Fischer baby monster F2 in [24].The purpose of this note is to prove the following Theorem. The conjecture of Frobenius is true for all the sporadic simple groups.The proof of our theorem has been carried out in the following way with the use of a computer. Let G be one of the sporadic simple groups. By [24] we may assume that G + Fx and G + F2. Let f(G, t) be the number of elements of order t in G and Ord(G) = (order of x \ x e G}. Tables of f(G, t) are given in the Appendix; see the supplements section at the end of this issue. For f(G,t) the reader is referred to the following papers: We look for all the subsets ß of Ord(G) satisfying the following two conditions: (a) If t is a member of ß, then ß contains all divisors of t. In particular 1 is always a member of ß.(b) For the subset ß of Ord(G) in (a), E,eS2/(G, /) divides |G|.A PASCAL program effectively generates such a subset ß with the use of a recursive concept. Special add and divide routines are used for explicit calculation with very large digit numbers. Then we have only six possibilities for ß:(i) ß = {1}, (ii) ß = Ord(G), (iii) ß = (1,2,3,4,5,10,11} and G is the Mathieu group MX2, (iv) ß = (1,2,3,5,7,11,19} and G is the Janko group Jx,

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The simple group of order 604,800

Cited by 58 publications

References 12 publications

Finite linear groups of degree seven. II

Finite linear groups of degree seven. II

Uniqueness of the graph of a rank three group

A conjecture of Frobenius and the sporadic simple groups. II

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