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Concept

Normal subgroup

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^{-1} \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup o

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Structural Properties Of Dendrite Groups

Nicolas Monod

Let G be the homeomorphism group of a dendrite. We study the normal subgroups of G. For instance, there are uncountably many nonisomorphic such groups G that are simple groups. Moreover, these groups can be chosen so that any isometric G-action on any metric space has a bounded orbit. In particular they have the fixed point property (FH).

AMER MATHEMATICAL SOC2019

Special Reductive Groups Over An Arbitrary Field

Mathieu Huruguen

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type, and special quasisplit reductive groups over an arbitrary field k. Finally, we give an application to a conjecture of Serre.

Springer Birkhauser2016

Let K be an algebraically closed field of characteristic

p\geq0

and let

Y=SPin_{2n+1}(K) (n\geq3)

be a simply connected simple algebraic group of type

B_n

over

K

. Also let

X

be the subgroup of type

D_n

, embedded in

Y

in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for

Y

. In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational

KY

-modules on which

X

acts with exactly two composition factors, completing the work of Ford in [12].

2017

MATH-310: Algebra

Study basic concepts of modern algebra: groups, rings, fields.

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L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux de ce sujet.

COM-501: Advanced cryptography

This course reviews some failure cases in public-key cryptography. It introduces some cryptanalysis techniques. It also presents fundamentals in cryptography such as interactive proofs. Finally, it presents some techniques to validate the security of cryptographic primitives.

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In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fi

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are