Fundamental theorem on homomorphismsIn abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems. Given two groups G and H and a group homomorphism f : G → H, let N be a normal subgroup in G and φ the natural surjective homomorphism G → G/N (where G/N is the quotient group of G by N).
CosetIn mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint, equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H].
Social equalitySocial equality is a state of affairs in which all individuals within a specific society have equal rights, liberties, and status, possibly including civil rights, freedom of expression, autonomy, and equal access to certain public goods and social services. Social equality requires the absence of legally enforced social class or caste boundaries and the absence of discrimination motivated by an inalienable part of an individual's identity.
Borel equivalence relationIn mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X (in the product topology). Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤B F, if and only if there is a Borel function Θ : X → Y such that for all x,x' ∈ X, one has x E x' ⇔ Θ(x) F Θ(x').
Universal propertyIn mathematics, more specifically in , a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties.
Equivalence relationIn mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
Left communismLeft communism, or the communist left, is a position held by the left wing of communism, which criticises the political ideas and practices espoused by Marxist–Leninists and social democrats. Left communists assert positions which they regard as more authentically Marxist than the views of Marxism–Leninism espoused by the Communist International after its Bolshevization by Joseph Stalin and during its second congress. In general, there are two currents of left communism, namely the Italian and Dutch–German left.
Presentation of a groupIn mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.
Conspiracy theoryA conspiracy theory is an explanation for an event or situation that asserts the existence of a conspiracy by powerful and sinister groups, often political in motivation, when other explanations are more probable. The term generally has a negative connotation, implying that the appeal of a conspiracy theory is based in prejudice, emotional conviction, or insufficient evidence.
Far-left politicsFar-left politics, also known as the radical left or extreme left, are politics further to the left on the left–right political spectrum than the standard political left. The term does not have a single, coherent definition; some scholars consider it to represent the left of social democracy, while others limit it to the left of communist parties. In certain instances—especially in the news media—far left has been associated with some forms of authoritarianism, anarchism, communism, and Marxism, or are characterized as groups that advocate for revolutionary socialism and related communist ideologies, or anti-capitalism and anti-globalization.
New LeftThe New Left was a broad political movement mainly in the 1960s and 1970s. It consisted of activists in the Western world who campaigned for a broad range of social issues such as civil and political rights, feminism, gay rights, rejection of gender roles, and drug policy reforms. Some see the New Left as an oppositional reaction to earlier Marxist and labor union movements for social justice that focused on dialectical materialism and social class, while others who used the term see the movement as a continuation and revitalization of traditional leftist goals.
