The graph extension theorem

Author:
Ernest Shult

Journal:
Proc. Amer. Math. Soc. **33** (1972), 278-284

MSC:
Primary 20B99

DOI:
https://doi.org/10.1090/S0002-9939-1972-0294477-5

MathSciNet review:
0294477

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Abstract: A sufficient condition is given that a transitive permutation group *G* admits a transitive extension . The condition is graph-theoretic and does not involve any direct algebraic properties of the group being extended. The result accounts for a fairly wide class of doubly transitive groups, including the two doubly transitive representations of the groups , and the doubly transitive representations of the Higman-Sims group, and the Conway group (.3).

**[1]**J. H. Conway,*A group of order*8,315,553,613,086,720,000, Bull. London Math. Soc.**1**(1969), 79-88. MR**40**#1470. MR**0248216 (40:1470)****[2]**G. Higman,*On the simple group of D. G. Higman and C. C. Sims*, Illinois J. Math.**13**(1969), 74-80. MR**39**#1545. MR**0240193 (39:1545)****[3]**D. G. Higman,*Solvability of a class of rank*3*permutation groups*, Nagoya Math. J.**41**(1971), 89-96. MR**0276316 (43:2063)****[4]**J. McLaughlin,*A simple group of order*898,128,000, Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, 1969, pp. 109-111. MR**39**#4268. MR**0242941 (39:4268)****[5]**E. Shult,*Characterizations of certain classes of graphs*, J. Combinatorial Theory (to appear). MR**0311518 (47:80)****[6]**-,*Supplement to ``The graph extension theorem"*, University of Florida, Gainesville, Fla. (mimeographed notes).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0294477-5

Keywords:
Doubly transitive groups,
transitive extension,
automorphism groups of graphs

Article copyright:
© Copyright 1972
American Mathematical Society