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. That is, [math]\displaystyle{ G }[/math] is an absolutely simple group if the only serial subgroups of [math]\displaystyle{ G }[/math] are [math]\displaystyle{ \{ e \} }[/math] (the trivial subgroup), and [math]\displaystyle{ G }[/math] 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