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simple group

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simple group (plural simple groups)

1. (group theory) A group which has no normal subgroups apart from the trivial group and itself.
   • 1989, Roger W. Carter, Simple Groups of Lie Type, John Wiley & Sons, Wiley Classics Edition, page 1,

     Since the first edition of this book was published in 1972, great progress has been made in the theory of finite simple groups. Above all, the classification of the finite simple groups was finally completed in 1981. […] In addition, the five Mathieu groups have been supplemented by the discovery between 1965 and 1981 of 21 further 'sporadic' simple groups.

   • 2000, Michael Aschbacher, Finite Group Theory, 2nd edition, Cambridge University Press, page 260:

     Let ${\displaystyle {\mathcal {K}}}$ be the list of finite simple groups appearing in section 47. […]
     Classification Theorem. Every finite simple group is isomorphic to a member of ${\displaystyle {\mathcal {K}}}$.

   • 2010, John Stillwell, Mathematics and Its History, 3rd edition, Springer, page 495:

     However, classification of the finite simple groups was much harder than could have been foreseen in the 19th century. It turned out to be easier (though still very hard) to classify continuous simple groups. […] Each continuous simple group is the symmetry group of a space with hypercomplex coordinates, either from ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {C} }$, ${\displaystyle \mathbb {H} }$, or ${\displaystyle \mathbb {O} }$.

• Almost simple group on Wikipedia.Wikipedia
• Characteristically simple group on Wikipedia.Wikipedia
• Quasisimple group on Wikipedia.Wikipedia
• Semisimple group on Wikipedia.Wikipedia
• List of finite simple groups on Wikipedia.Wikipedia