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On Unsolvable Groups of Degree p = 4q + 1, p and q Primes

  • K. I. Appel (a1) and E. T. Parker (a1)

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This paper presents two results. They are:

Theorem 1. Let G be a doubly transitive permutation group of degree nq + 1 where a is a prime and n < g. If G is neither alternating nor symmetric, then G has Sylow q-subgroup of order only q.

Result 2. There is no unsolvable transitive permutation group of degree p = 29, 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree p.

Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.

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References

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1. Burnside, W., Theory of groups of finite order, 2nd ed. (Cambridge, 1911).
2. Hall, M. Jr., The theory of groups (New York, 1959).
3. Ito, N., Zur Theorie der Permutationsgruppen von Grad p, Math. Z., 74 (1960), 299301.
4. Miller, G. A., Limits of the degree of transitivity of substitution groups, Bull. Amer. Math Soc., 22 (1915), 6871.
5. Parker, E. T., On quadruply transitive groups, Pacific J. Math., 9 (1959), 829836.
6. Parker, E. T. and Nikolai, J., A search for analogues of the Mathieu groups, Math Comput., 12 (1958), 3843.
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On Unsolvable Groups of Degree p = 4q + 1, p and q Primes

  • K. I. Appel (a1) and E. T. Parker (a1)

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