O'Nan group
Encyclopedia
In the mathematical field of group theory
, the O'Nan group O'N is a sporadic
simple group
of order
It was found by in a study of groups with a Sylow 2-subgroup of "Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2nZ ×Z/2nZ ×Z/2nZ).PSL3(F2).
The Schur multiplier
has order 3, and its outer automorphism group has order 2. Its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism (Ryba 1988).
The subgroup fixed by an outer automorphism of order 2 is the Janko group J1
, a maximal subgroup.
O'N is one of the 6 sporadic simple groups called the pariahs
, because they are not found within the Monster group
.
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...
, the O'Nan group O'N is a sporadic
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
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...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
- 29 · 34 · 5 · 73 · 11 · 19 · 31
- = 460815505920
It was found by in a study of groups with a Sylow 2-subgroup of "Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2nZ ×Z/2nZ ×Z/2nZ).PSL3(F2).
The Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
has order 3, and its outer automorphism group has order 2. Its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism (Ryba 1988).
The subgroup fixed by an outer automorphism of order 2 is the Janko group J1
Janko group J1
In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century...
, a maximal subgroup.
O'N is one of the 6 sporadic simple groups called the pariahs
Pariah group
In mathematical group theory, the term pariah was introduced by to refer to the six sporadic simple groups that are not subquotients of the monster simple group.These groups are:*Three of the Janko groups: J1, J3, and J4.*The Lyons group...
, because they are not found within the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
.