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 subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.[1]
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).[2][3]
If H is a subgroup of G, then G is sometimes called an overgroup of H.
The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.
as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group...
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...
commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important...
Parabolic subgroup may refer to: a parabolic subgroup of a reflection group a subgroup of an algebraic group that contains a Borel subgroup This disambiguation...
S\subseteq G} fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers...
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...
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...
group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it...
A language family is a group of languages related through descent from a common ancestral language or parental language, called the proto-language of that...
{\displaystyle p} , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup of G {\displaystyle...
Serpentine subgroup (part of the kaolinite-serpentine group in the category of phyllosilicates) are greenish, brownish, or spotted minerals commonly found...
In mathematics, the lattice of subgroups of a group G {\displaystyle G} is the lattice whose elements are the subgroups of G {\displaystyle G} , with the...
said to have no small subgroup if there exists a neighborhood U {\displaystyle U} of the identity that contains no nontrivial subgroup of G . {\displaystyle...
Subgroup analysis refers to repeating the analysis of a study within subgroups of subjects defined by a subgrouping variable. For example: smoking status...
subgroup of G {\displaystyle G} admits a unique smooth structure which makes it an embedded Lie subgroup of G {\displaystyle G} —i.e. a Lie subgroup such...
group theory, for a given group G, the diagonal subgroup of the n-fold direct product G n is the subgroup { ( g , … , g ) ∈ G n : g ∈ G } . {\displaystyle...
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...
their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder...
In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of...
algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general...
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...
fruits of one of a number of banana cultivars belonging to the Cavendish subgroup of the AAA banana cultivar group (triploid cultivars of Musa acuminata)...
under multiplication. Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums...
group G, one can form the subgroup that consists of all its integer powers: ⟨g⟩ = { gk | k ∈ Z }, called the cyclic subgroup generated by g. The order...
taken to be a subgroup defined in terms of a prime number p, usually the local subgroups, the normalizers of the nontrivial p-subgroups. In which case...
when talking about subgroups of G . {\displaystyle G.} The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same...
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...
solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof...
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...
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