mathematics, specifically group theory, a subgroup series of a group G {\displaystyle G} is a chain of subgroups: 1 = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G {\displaystyle...

a composition series is a maximal subnormal series, while a chief series is a maximal normal series. If a group G has a normal subgroup N, then the factor...

commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...

the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator...

normal subgroup of G/Ni. Equivalently, there does not exist any subgroup A normal in G such that Ni < A < Ni+1 for any i. In other words, a chief series may...

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...

Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory...

for any pair g, h ∈ G. ascendant subgroup A subgroup H of a group G is ascendant if there is an ascending subgroup series starting from H and ending at G...

investigations of nilpotent normal subgroups. A Fitting chain (or Fitting series or nilpotent series) for a group is a subnormal series with nilpotent quotients...

Theory. Krieger, 1975. Schmidt, Roland. Subgroup Lattices of Groups. de Gruyter, 1994. Keith Conrad, SUBGROUP SERIES II, Section 4 , http://www.math.uconn...

Serpentine subgroup (part of the kaolinite-serpentine group in the category of phyllosilicates) are greenish, brownish, or spotted minerals commonly found...

is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group. For a group...

definitions for a nilpotent group G: G has a central series of finite length. That is, a series of normal subgroups { 1 } = G 0 ◃ G 1 ◃ ⋯ ◃ G n = G {\displaystyle...

subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of...

return the minimal word, but it can return optimal words based on the series of subgroups that is used. The code looks like this: function operate(element...

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...

an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever...

In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group. Let G {\displaystyle...

the Puig subgroup, introduced by Puig (1976), is a characteristic subgroup of a p-group analogous to the Thompson subgroup. If H is a subgroup of a group...

2016. Retrieved January 30, 2017. "John Glenn Archives, Audiovisuals Subgroup, Series 3: Certificates". Ohio State University. Archived from the original...

spelled backwards, and the agemo subgroup is denoted by an upside-down omega (℧). The omega subgroups are the series of subgroups of a finite p-group, G, indexed...

groups, a Grosshans subgroup, named after Frank Grosshans, is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring...

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...

(or most) of the unitary dual could be parametrised by starting with a subgroup H of G, simpler but not compact, and building up induced representations...

reflection groups, the parabolic subgroups are a special kind of subgroup. The precise definition of which subgroups are parabolic depends on context—for...

In algebraic topology, a peripheral subgroup for a space-subspace pair X ⊃ Y is a certain subgroup of the fundamental group of the complementary space...

In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory. It may be seen as a generalisation of Baker's...

every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup H {\displaystyle...

abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced...

the upper central series. Generalizing the earlier comments about the socle, a finite p-group with order pn contains normal subgroups of order pi with...