Absolutely simple group

In mathematics, in the field of group theory, a group is said to be absolutely simple if it has no proper nontrivial serial subgroups.[1] That is, ${\displaystyle G}$ is an absolutely simple group if the only serial subgroups of ${\displaystyle G}$ are ${\displaystyle \{e\}}$ (the trivial subgroup), and ${\displaystyle G}$ itself (the whole group).

In the finite case, a group is absolutely simple if and only if it is simple. However, in the infinite case, absolutely simple is a stronger property than simple. The property of being strictly simple is somewhere in between.

See also

• Ascendant subgroup
• Strictly simple group

References

1. Robinson, Derek J. S. (1996), A course in the theory of groups, Graduate Texts in Mathematics, 80 (Second ed.), New York: Springer-Verlag, p. 381, doi:10.1007/978-1-4419-8594-1, ISBN 0-387-94461-3, MR 1357169.