sekundaariryhmien

Sekundaariryhmien, also known as secondary groups, are a concept in group theory, a branch of abstract algebra. They are used to classify finite groups and are particularly useful in the study of finite simple groups. A secondary group is defined as a group that is a central extension of a finite group by a cyclic group of prime order. This means that the group has a normal subgroup that is cyclic of prime order, and the quotient group by this subgroup is isomorphic to the original finite group.

Secondary groups are important because they provide a way to construct new groups from existing ones. Given a finite group G and a prime p, one can construct a secondary group by taking the central product of G with a cyclic group of order p. This construction is useful in the classification of finite simple groups, as it allows one to reduce the problem of classifying all finite simple groups to the problem of classifying all finite simple groups that are not secondary groups.

The study of secondary groups also has applications in other areas of mathematics, such as representation theory and modular representation theory. In representation theory, secondary groups are used to study the representations of finite groups over fields of prime characteristic. In modular representation theory, secondary groups are used to study the representations of finite groups over fields of characteristic p, where p is a prime number.

In summary, secondary groups are a concept in group theory that are used to classify finite groups and have applications in other areas of mathematics. They are defined as groups that are central extensions of a finite group by a cyclic group of prime order, and they provide a way to construct new groups from existing ones. The study of secondary groups is an active area of research in mathematics.