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Totally Variant Sets in Finite Groups and Vector Spaces

Published online by Cambridge University Press:  20 November 2018

Frederick Hoffman  and
Lloyd R. Welch
Frederick Hoffman
Affiliation:
Institute for Defense Analyses, Princeton, N.J.
Lloyd R. Welch
Affiliation:
Institute for Defense Analyses, Princeton, N.J.
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We are concerned here with the question of which finite groups and vector spaces possess subsets which are moved by every non-identity automorphism (in the vector-space case—non-singular linear transformation). We find that this is the case for all but four finite-dimensional vector spaces (2-, 3-, and 4-dimensional space over Z2, 2-dimensional space over Z3), and for all finite groups except for those corresponding to the vector-space exceptions, and the quaternion group of order eight. The question was first posed to the authors, in the vector-space case, by Morris Marx.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 701 - 710
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Hall, M., The theory of groups (Macmillan, New York, 1959).Google Scholar
2. Hall, M. and Senior, J. K., The groups of order 2n (n < 6) (Macmillan, New York, 1964).Google Scholar