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 characteristically simple if it has no proper nontrivial
characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...
s. Characteristically simple groups are sometimes also termed elementary groups. Characteristically simple is a ''weaker'' condition than being a
simple group
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The d ...
, as simple groups must not have any proper nontrivial
normal subgroups, which include characteristic subgroups.
A finite group is characteristically simple if and only if it is the
direct product of
isomorphic simple groups. In particular, a finite
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
is characteristically simple if and only if it is an
elementary abelian group
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...
. This does not hold in general for
infinite group
In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order.
Examples
* (Z, +), the group of integers with addition is infi ...
s; for example, the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s form a characteristically simple group that is not a direct product of simple groups.
A minimal normal subgroup of a group ''G'' is a nontrivial normal subgroup ''N'' of ''G'' such that the only proper subgroup of ''N'' that is normal in ''G'' is the trivial subgroup. Every minimal normal subgroup of a group is characteristically simple. This follows from the fact that a characteristic subgroup of a normal subgroup is normal.
References
*
Properties of groups
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