In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
of a group
is termed seminormal if there is a subgroup
such that
, and for any proper subgroup
of
,
is a proper subgroup of
.
This definition of seminormal subgroups is due to
Xiang Ying Su.
[. Foguel writes: "Su introduces the concept of seminormal subgroups and using this tool he gives four sufficient conditions for supersolvability."]
Every
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
is seminormal. For finite groups, every
quasinormal subgroup is seminormal.
References
{{reflist
Subgroup properties