Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1 or e depending on the context. If the group operation is denoted ∗ then it is defined by ee = e.

The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group.

The trivial group should not be confused with the empty set (which has no elements, and lacking an identity element, cannot be a group).

Given any group G, the group consisting of only the identity element is a subgroup of G, and, being the trivial group, is called the trivial subgroup of G.

The term, when referred to "G has no nontrivial proper subgroups" refers to the only subgroups of G being the trivial group {e} and the group G itself.

Properties

The trivial group is cyclic of order 1; as such it may be denoted Z1 or C1. If the group operation is called addition, the trivial group is usually denoted by 0. If the group operation is called multiplication then 1 can be a notation for the trivial group.

The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.

See also

References

  • Rowland, Todd and Weisstein, Eric W. "Trivial Group". MathWorld.CS1 maint: Multiple names: authors list (link)
0G

0G, 0-G, or 0/G may refer to:

0G, or Zero Generation is the mobile telephony for Mobile radio telephone

0-G, or Zero gravity is the absence of g-force, also called Weightlessness

Zero-g roll, one of the common Roller coaster elements

Zero Gravity Corporation

0/G, a model of Ν-Asurada AKF-0

Zero game, a state in game theory where neither player has any legal options

Zero grade, a type of Indo-European ablaut

Zero group, a type of mathematical Trivial group

0G, a size of plugs, for ears (8mm)

Andrews–Curtis conjecture

In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not.

It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.

Conjugate closure

In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the set of the conjugates of the elements of S:

SG = {g−1sg | gG and sS}

The conjugate closure of S is denoted <SG> or <S>G.

The conjugate closure of any subset S of a group G is always a normal subgroup of G; in fact, it is the smallest (by inclusion) normal subgroup of G which contains S. For this reason, the conjugate closure is also called the normal closure of S or the normal subgroup generated by S. The normal closure can also be characterized as the intersection of all normal subgroups of G which contain S. Any normal subgroup is equal to its normal closure.

The conjugate closure of a singleton subset {a} of a group G is a normal subgroup generated by a and all elements of G which are conjugate to a. Therefore, any simple group is the conjugate closure of any non-identity group element. The conjugate closure of the empty set is the trivial group.

Contrast the normal closure of S with the normalizer of S, which is (for S a group) the largest subgroup of G in which S itself is normal. (This need not be normal in the larger group G, just as <S> need not be normal in its conjugate/normal closure.)

Dual to the concept of normal closure is that of normal interior or normal core, defined as the join of all normal subgroups contained in S.

E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases.

The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56.

F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.

The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.

The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E8.

In older books and papers, F4 is sometimes denoted by E4.

Frattini subgroup

In mathematics, the Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group e or the Prüfer group, it is defined by Φ(G) = G. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below). It is named after Giovanni Frattini, who defined the concept in a paper published in 1885.

Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, and such that every element's expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis {1}. Addition of integers is commutative, associative, and has subtraction as its inverse operation, each integer is the sum or difference of some number of copies of the number 1, and each integer has a unique representation as an integer multiple of the number 1.

Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.

The elements of a free abelian group with basis B may be described in several equivalent ways. These include formal sums over B, expressions of the form where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, and the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions.

Every set B has a free abelian group with B as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B. Alternatively, the free abelian group with basis B may be described by a presentation with the elements of B as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.

Free product

In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group GH. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the most general group that can be made from a given set of generators).

The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups.

The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces. In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is simply the free product of the fundamental groups of the spaces.

Free products are also important in Bass–Serre theory, the study of groups acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to the free product of cyclic groups of orders 4 and 6 amalgamated over a cyclic group of order 2.

The free product (= coproduct) of groups is nicely set in the context of Categories and Groupoids in Philip Higgins 1971 book referenced below. The point is that a disjoint union of groups is not a group but it is a groupoid. A groupoid has a universal group and the universal group of a disjoint union of groups is the free product (= coproduct) of the groups.

Generating set of a group

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

In other words, if S is a subset of a group G, then ⟨S⟩, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, ⟨S⟩ is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Notice: here inverses are only needed if the group is infinite. For a finite group, the inverse can be expressed as a power of the element itself.)

If G = ⟨S⟩, then we say that S generates G, and the elements in S are called generators or group generators. If S is the empty set, then ⟨S⟩ is the trivial group {e}, since we consider the empty product to be the identity.

When there is only a single element x in S, ⟨S⟩ is usually written as ⟨x⟩. In this case, ⟨x⟩ is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that ⟨x⟩ equals the entire group G. For finite groups, it is also equivalent to saying that x has order |G|.

If G is a topological group then a subset S of G is called a set of topological generators if ⟨S⟩ is dense in G i.e. the closure of ⟨S⟩ is the whole group G.

Identity function

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x.

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses:

The order of a group is its cardinality, i.e., the number of elements in its set. Also, the order, sometimes period, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order.

The ordering relation of a partially or totally ordered group.This article is about the first sense of order.

The order of a group G is denoted by ord(G) or |G| and the order of an element a is denoted by ord(a) or |a|.

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

The binary operation of a semigroup is most often denoted multiplicatively: x·y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z) for all x, y and z in the semigroup.

The name "semigroup" originates in the fact that a semigroup generalizes a group by preserving only associativity and closure under the binary operation from the axioms defining a group while omitting the requirement for an identity element and inverses. From the opposite point of view (of adding rather than removing axioms), a semigroup is an associative magma. As in the case of groups or magmas, the semigroup operation need not be commutative, so x·y is not necessarily equal to y·x; a typical example of associative but non-commutative operation is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called an abelian semigroup.

A monoid is an algebraic structure intermediate between groups and semigroups, and is a semigroup having an identity element, thus obeying all but one of the axioms of a group; existence of inverses is not required of a monoid. A natural example is strings with concatenation as the binary operation, and the empty string as the identity element. Restricting to non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form a commutative semigroup that is not a monoid, whereas the non-negative integers do form a monoid. A semigroup without an identity element can be easily turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups should not be confused with quasigroups, which are a generalization of groups in a different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups a notion of division. Division in semigroups (or in monoids) is not possible in general.

The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semigroups realizing any semigroup as transformation semigroup, in which arbitrary functions replace the role of bijections from group theory. Other fundamental techniques of studying semigroups like Green's relations do not imitate anything in group theory though. A deep result in the classification of finite semigroups is Krohn–Rhodes theory. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov processes. In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time. There are numerous special classes of semigroups, semigroups with additional properties, which appear in particular applications. Some of these classes are even closer to groups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups, orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There also interesting classes of semigroups that do not contain any groups except the trivial group; examples of the latter kind are bands and their commutative subclass—semilattices, which are also ordered algebraic structures.

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

The complete classification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics.

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

Special unitary group

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n×n unitary matrices with determinant 1.

(More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.)

The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on Cn. It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ GL(n, C).

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Sporadic group

In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions that do not follow such a systematic pattern. These are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case the sporadic groups number 27.

The monster group is the largest of the sporadic groups and contains all but six of the other sporadic groups as subgroups or subquotients.

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 subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually 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.

A proper subgroup of a group G is a subgroup H which is a proper subset of G (i. e. H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (i. e. {e} ≠ H ≠ G).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. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.

This article will write ab for a ∗ b, as is usual.

Zero object (algebra)

In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism).

Instances of the zero object include, but are not limited to the following:

These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties.

In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as:

κ0 = 0 , where κR.

The most general of them, the zero module, is a finitely-generated module with an empty generating set.

For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R,

In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see the section Unital structures below.

Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module.

The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra.

The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.

Zero ring

In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which xy = 0 for all x and y. This article refers to the one-element ring.)

In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object.

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