In
mathematics, in the field of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
is said to be strictly simple if it has no proper nontrivial
ascendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its successo ...
s. That is,
is a strictly simple group if the only ascendant subgroups of
are
(the trivial subgroup), and
itself (the whole group).
In the finite case, a group is strictly simple if and only if it is
simple
Simple or SIMPLE may refer to:
* Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by John ...
. However, in the infinite case, strictly simple is a stronger property than simple.
See also
*
Serial subgroup
In the mathematical field of group theory, a subgroup ''H'' of a given group ''G'' is a serial subgroup of ''G'' if there is a chain ''C'' of subgroups of ''G'' extending from ''H'' to ''G'' such that for consecutive subgroups ''X'' and ''Y'' in ' ...
*
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, G is an absolutely simple group if the only serial subgroups of G are \ (the trivial subgroup), and G ...
ReferencesSimple Group
Encyclopedia of Mathematics, retrieved 1 January 2012
Properties of groups
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