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Finite Groups Which Admit An Automorphism With Few Orbits

  • Daniel Gorenstein (a1)

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In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.

In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

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References

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1. Burnside, W., Theory of groups of finite order (Dover, New York, 1955).
2. Feit, W., On the structure of Frobenius groups, Can. J. Math., 9 (1957), 587–96.
3. Gorenstein, D., A class of Frobenius groups, Can. J. Math., 11 (1959), 3942.
4. Gorenstein, D. and Herstein, I.N., A class of solvable groups, Can. J. Math., 11 (1959), 311-20.
5. Higman, G., Groups and rings which have automorphisms without non-trivial fixed elements J. London Math. Soc, 32 (1957), 321334.
6. Zassenhaus, H., The theory of groups (Chelsea, New York, 1958).
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Finite Groups Which Admit An Automorphism With Few Orbits

  • Daniel Gorenstein (a1)

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