Critical group

In mathematics, in the realm of group theory, a group is said to be critical if it is not in the variety generated by all its proper subquotients, which includes all its subgroups and all its quotients.^[1]

Any finite monolithic A-group is critical. This result is due to Kovacs and Newman.^[2] But not every monolithic group is critical.^[3]
The variety generated by a finite group has a finite number of nonisomorphic critical groups.^[1]

References

^ ^a ^b Oates, Sheila; Powell, M.B (April 1964). "Identical relations in finite groups" (PDF). Journal of Algebra. 1 (1): 11–39. doi:10.1016/0021-8693(64)90004-3. Retrieved 26 April 2024.
^ Kovács, L. G.; Newman, M. F. (May 1966). "On critical groups". Journal of the Australian Mathematical Society. 6 (2): 237–250. doi:10.1017/S144678870000481X.
^ Neumann, Hanna (6 December 2012). Varieties of Groups. Springer Science & Business Media. p. 147. ISBN 978-3-642-88599-0.

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