Lattice of subgroupsIn mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection. The dihedral group Dih4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four.
Permutation groupIn mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by Sn, and may be called the symmetric group on n letters.
Residually finite groupIn the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group, such that There are a number of equivalent definitions: A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element. A group is residually finite if and only if the intersection of all its subgroups of finite index is trivial.
Type systemIn computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type (for example, integer, floating point, string) to every "term" (a word, phrase, or other set of symbols). Usually the terms are various constructs of a computer program, such as variables, expressions, functions, or modules. A type system dictates the operations that can be performed on a term. For variables, the type system determines the allowed values of that term.
EmailElectronic mail (email or e-mail) is a method of transmitting and receiving messages using electronic devices. It was conceived in the late–20th century as the digital version of, or counterpart to, mail (hence e- + mail). Email is a ubiquitous and very widely used communication medium; in current use, an email address is often treated as a basic and necessary part of many processes in business, commerce, government, education, entertainment, and other spheres of daily life in most countries.
Type safetyIn computer science, type safety and type soundness are the extent to which a programming language discourages or prevents type errors. Type safety is sometimes alternatively considered to be a property of facilities of a computer language; that is, some facilities are type-safe and their usage will not result in type errors, while other facilities in the same language may be type-unsafe and a program using them may encounter type errors.
Inner automorphismIn abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.
Email spoofingEmail spoofing is the creation of email messages with a forged sender address. The term applies to email purporting to be from an address which is not actually the sender's; mail sent in reply to that address may bounce or be delivered to an unrelated party whose identity has been faked. Disposable email address or "masked" email is a different topic, providing a masked email address that is not the user's normal address, which is not disclosed (for example, so that it cannot be harvested), but forwards mail sent to it to the user's real address.
Automorphism groupIn mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group.
Data typeIn computer science and computer programming, a data type (or simply type) is a collection or grouping of data values, usually specified by a set of possible values, a set of allowed operations on these values, and/or a representation of these values as machine types. A data type specification in a program constrains the possible values that an expression, such as a variable or a function call, might take. On literal data, it tells the compiler or interpreter how the programmer intends to use the data.
Dependent typeIn computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.
Email addressAn email address identifies an email box to which messages are delivered. While early messaging systems used a variety of formats for addressing, today, email addresses follow a set of specific rules originally standardized by the Internet Engineering Task Force (IETF) in the 1980s, and updated by . The term email address in this article refers to just the addr-spec in Section 3.4 of RFC 5322. The RFC defines address more broadly as either a mailbox or group.
Type conversionIn computer science, type conversion, type casting, type coercion, and type juggling are different ways of changing an expression from one data type to another. An example would be the conversion of an integer value into a floating point value or its textual representation as a string, and vice versa. Type conversions can take advantage of certain features of type hierarchies or data representations.
Email spamEmail spam, also referred to as junk email, spam mail, or simply spam, is unsolicited messages sent in bulk by email (spamming). The name comes from a Monty Python sketch in which the name of the canned pork product Spam is ubiquitous, unavoidable, and repetitive. Email spam has steadily grown since the early 1990s, and by 2014 was estimated to account for around 90% of total email traffic. Since the expense of the spam is borne mostly by the recipient, it is effectively postage due advertising.
Profinite groupIn mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists such that every group in the system can be generated by elements.
Congruence subgroupIn mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.
AutomorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space.
Intuitionistic type theoryIntuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions.
Email encryptionEmail encryption is encryption of email messages to protect the content from being read by entities other than the intended recipients. Email encryption may also include authentication. Email is prone to the disclosure of information. Most emails are encrypted during transmission, but they are stored in clear text, making them readable by third parties such as email providers. By default, popular email services such as Gmail and Outlook do not enable end-to-end encryption.
Group actionIn mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it.
