Characteristically simple group
Encyclopedia
In mathematics
, in the field of group theory
, a group
is said to be characteristically simple if it has no proper nontrivial characteristic subgroup
s. Characteristically simple groups are sometimes also termed elementary groups. Characteristically simple is a weaker condition than being a simple group
, 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
is characteristically simple if and only if it is an elementary abelian group
. This does not hold in general for infinite groups; for example, the rational number
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is said to be characteristically simple if it has no proper nontrivial characteristic subgroup
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
s. Characteristically simple groups are sometimes also termed elementary groups. Characteristically simple is a weaker condition than being a simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
, 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
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of isomorphic simple groups. In particular, a finite solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
is characteristically simple if and only if it is an elementary abelian group
Elementary Abelian group
In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a p-group....
. This does not hold in general for infinite groups; for example, the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
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
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
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
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
of a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
is normal.