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Characteristically simple groups are sometimes also termed elementary groups.
This does not hold in general for infinite groups; for example, the rational numbers form a characteristically simple group that is not a direct product of simple groups.
Advances on the global structure of totally disconnected groups have been obtained in 2011 by Caprace and Monod, with notably a clasification of characteristically simple groups and of Noetherian groups.
Wörterbuch für Mathematik