In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the realm 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 capable if it occurs as the
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
group of some group. These groups were first studied by
Reinhold Baer
Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups.
Biography
Baer studied mechanical engineering f ...
, who showed that a finite
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
is capable if and only if it is a product of cyclic groups of orders ''n''
1,...,''n''
''k'' where ''n''
''i'' divides ''n''
''i''+1 and ''n''
''k''–1=''n''
''k''.
References
*
External links
Bounds on the index of the center in capable groups
Properties of groups
{{Abstract-algebra-stub