Conjugate-permutable subgroup
In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997 and arose in
Component (group theory)
In mathematics, in the field of group theory, a component of a finite group is a quasisimple subnormal subgroup. Any two distinct components commute. The product of all the components is the layer of
C-normal subgroup
In mathematics, in the field of group theory, a subgroup of a group is called c-normal if there is a normal subgroup of such that and the intersection of and lies inside the normal core of . For a wea
Ascendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in
Pure subgroup
In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and relat
Conjugacy-closed subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be conjugacy-closed if any two elements of the subgroup that are conjugate in the group are also conjugate in the subgrou
Quasinormal subgroup
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of
Contranormal subgroup
In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whosenormal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is th
Malnormal subgroup
In mathematics, in the field of group theory, a subgroup of a group is termed malnormal if for any in but not in , and intersect in the identity element. Some facts about malnormality:
* An intersect
Special abelian subgroup
In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup. Equival
Weakly normal subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be weakly normal if whenever , we have . Every pronormal subgroup is weakly normal.
Maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no
Seminormal subgroup
In mathematics, in the field of group theory, a subgroup 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 o
Fully normalized subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be fully normalized if every automorphism of the subgroup lifts to an inner automorphism of the whole group. Another way
Polynormal subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugatio
Essential subgroup
In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept
Basic subgroup
In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulik
CEP subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to have the Congruence Extension Property or to be a CEP subgroup if every congruence on the subgroup lifts to a congruence
Paranormal subgroup
In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within t
Retract (group theory)
In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup.
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because
Subnormal subgroup
In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning
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.
Central subgroup
In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group. Given a group , the center of , denoted as , is defined as the set of t
Centrally-closed subgroup
In mathematics, in the realm of group theory, a subgroup of a group is said to be centrally closed if the centralizer of any nonidentity element of the subgroup lies inside the subgroup. Some facts ab
Carter subgroup
In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, a
Semipermutable subgroup
In mathematics, in algebra, in the realm of group theory, a subgroup of a finite group is said to be semipermutable if commutes with every subgroup whose order is relatively prime to that of . Clearly
Transitively normal subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbol
Abnormal subgroup
In mathematics, specifically group theory, an abnormal subgroup is a subgroup H of a group G such that for all x in G, x lies in the subgroup generated by H and H x, where H x denotes the conjugate su
Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a su
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is
Pronormal subgroup
In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and a
Modular subgroup
In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the jo
Hall subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall.
Descendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in
Verbal subgroup
In mathematics, in the area of abstract algebra known as group theory, a verbal subgroup is a subgroup of a group that is generated by all elements that can be formed by substituting group elements fo
Serial subgroup
In the mathematical field of group theory, a subgroup H of a given group G is a serial subgroup of G if there is a chain C of subgroups of G extending from H to G such that for consecutive subgroups X