Subgroup properties

Subgroup properties in group theory refer to certain characteristics or conditions that a subgroup of a given group may satisfy. These properties help in categorizing subgroups and understanding their structure relative to the larger group.

In group theory, an **abnormal subgroup** is a specific type of subgroup that captures certain properties related to the structure of the group. A subgroup \( H \) of a group \( G \) is called **abnormal** if it satisfies the following condition: For every \( g \in G \), if \( gH \) (the left coset of \( H \) in \( G \)) intersects with \( H \) non-trivially (i.e.

In group theory, a branch of abstract algebra, an **ascendant subgroup** of a group \( G \) is a specific type of subgroup that has a unique property concerning its relation to the whole group.

A **C-normal subgroup** is a concept from group theory, a branch of mathematics that studies the algebraic structures known as groups. A subgroup \( N \) of a group \( G \) is termed a **C-normal subgroup** if it satisfies certain conditions related to its normality.

In the context of group theory, a **Carter subgroup** is a specific type of subgroup associated with a finite group, particularly in the study of nilpotent and solvable groups. Specifically, a Carter subgroup is defined as follows: - It is a subgroup that is the intersection of all Sylow subgroups corresponding to its normalizer in the group.

In group theory, a branch of abstract algebra, a **central subgroup** refers to a subgroup that is contained in the center of a given group. The center of a group \( G \), denoted \( Z(G) \), is defined as the set of all elements \( z \in G \) such that \( zg = gz \) for all \( g \in G \). In other words, the center consists of all elements that commute with every other element in the group.

In group theory, a subgroup \( H \) of a group \( G \) is said to be **centrally closed** if it is closed under the operation of conjugation by elements of the center of \( G \).

A **characteristic subgroup** of a group \( G \) is a subgroup \( H \) that is invariant under all automorphisms of the group \( G \). This means that for any automorphism \( \phi \) of \( G \), the image \( \phi(H) \) is still a subgroup of \( G \) and is equal to \( H \) itself.

In group theory, a **conjugate-permutable subgroup** is a specific type of subgroup that has a particular property related to conjugation. A subgroup \( H \) of a group \( G \) is said to be conjugate-permutable if for every element \( g \in G \), the following condition holds: \[ H^g = gHg^{-1} \text{ satisfies } H^g \cap H \neq \emptyset.

In group theory, a **contranormal subgroup** is a type of subgroup with a particular relationship to normal subgroups and normality conditions in a larger group.

In group theory, a branch of abstract algebra, the term "descendant subgroup" refers to a subgroup that is generated by certain elements of a group and is contained within a larger structure, typically in the context of the subgroup lattice.

In the context of group theory, the concept of a **fully normalized subgroup** pertains to a subgroup that is maximal with respect to the property of being normal in a certain sense. Specifically, a subgroup \( H \) of a group \( G \) is said to be fully normalized if it is normal in every subgroup of \( G \) that contains it.

A Hall subgroup is a concept from group theory, specifically in the study of finite groups. It is named after Philip Hall, who introduced the concept in his work on groups and combinatorics.

A **malnormal subgroup** is a specific type of subgroup within group theory, particularly in the context of group actions and normal subgroups.

In group theory, a branch of abstract algebra, a **maximal subgroup** is a specific type of subgroup of a given group. A subgroup \( M \) of a group \( G \) is called a maximal subgroup if it is proper (meaning that it is not equal to \( G \)) and is not contained in any other proper subgroup of \( G \). In other words, there are no subgroups \( N \) such that \( M < N < G \).

In the context of group theory, particularly in the study of modular lattices and modular subgroups, a **modular subgroup** is a specific type of subgroup that satisfies the modular law.

A **normal subgroup** is a special type of subgroup in the context of group theory, which is a branch of abstract algebra. Let's define it more precisely. Given a group \( G \) and a subgroup \( N \) of \( G \): 1. **Subgroup**: A subgroup \( N \) must itself be a group under the operation defined on \( G \).

A **paranormal subgroup** is a concept in group theory, specifically in the area of finite group theory. A subgroup \( H \) of a group \( G \) is said to be paranormal if it meets a specific condition related to its normality and the structure of \( G \).

A **polynormal subgroup** is a concept from group theory, particularly in the study of group extensions and solvable groups. A subgroup \( N \) of a group \( G \) is called **polynormal** if for every finite sequence of subgroups \( H_1, H_2, \ldots, H_n \) of \( G \) such that: 1. \( H_1 \) is a subgroup of \( N \), 2.

A pronormal subgroup is a specific type of subgroup in group theory, particularly in the context of finite groups. A subgroup \( H \) of a group \( G \) is said to be **pronormal** if, for every \( g \in G \), the intersection of \( H \) with \( H^g \) (the conjugate of \( H \) by \( g \)) is a normal subgroup of \( H \).

A **quasinormal subgroup** is a concept in group theory, a branch of abstract algebra. A subgroup \( H \) of a group \( G \) is said to be quasinormal if it is permutable with every subgroup of \( G \).

In group theory, a **seminormal subgroup** is a particular type of subgroup within a group that is related to the concept of normality.

A semipermutable subgroup is a concept in the field of group theory, particularly in the study of group extensions and solvable groups. A subgroup \( H \) of a group \( G \) is called **semipermutable** if for every normal subgroup \( N \) of \( G \) such that \( N \) is a subset of \( H \), the subgroup \( H \) permutes with \( N \) in \( G \).

In group theory, a branch of abstract algebra, a **serial subgroup** refers to certain kinds of normal subgroups within a group. Specifically, a subgroup \( H \) of a group \( G \) is termed a serial subgroup if it can be expressed in a specific way in relation to the entire group \( G \) and to other subgroups.

In the context of group theory, a **special abelian subgroup** usually refers to a specific type of subgroup within a group, particularly in the theory of finite groups or in the study of Lie algebras.

In group theory, a **subnormal subgroup** is a specific type of subgroup that has a particular relationship with the larger group it is part of.

In group theory, a subgroup \( H \) of a group \( G \) is said to be **transitively normal** in \( G \) if it is normal in every subgroup of \( G \) that contains \( H \).