Easy group (math)

A simple group is a mathematical object of algebra that is particularly considered in group theory .

Each group has itself and the set containing only the neutral element as its normal divisor . This raises the question of which groups have no other normal subdivisions. By definition, these are the simple groups.

definition

A group is called simple if it only has and with the neutral element as the normal divisor . In addition, it is required that one can say more succinctly: A group is called simple if it has exactly two normal factors. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \{e\}}$ ${\displaystyle e}$ ${\displaystyle G\neq \{e\}}$

Finite easy groups

→ Main article : Finite simple group

In group theory, finite simple groups are considered to be the “basic building blocks” of finite groups , since every finite group can be constructed from simple groups in a finite number of steps. Since 1982 the finite simple groups have been fully classified, the list consists of

• the cyclic groups of prime order,
• the alternating groups with , ${\displaystyle A_{n}}$ ${\displaystyle n\geq 5}$
• the Lie-type groups (16 infinite series each) and
• 26 sporadic groups .

Infinite simple groups

Infinite simple groups are not Abelian.

Examples

• The infinite alternating group , that is, the group of finite even permutations of natural numbers, is simple. This group can be constructed as a direct limit of all under the standard embeddings. ${\displaystyle A_{\infty }}$ ${\displaystyle A_{n}}$ ${\displaystyle A_{n}\to A_{n+1}}$
• Each group different from the two-element group with exactly two conjugation classes is an infinite simple group. ${\displaystyle C_{2}}$

Simple lie groups

Deviating from the above definition customary in group theory, in the theory of Lie groups (not to be confused with the above groups of the Lie type) a connected Lie group is called a simple Lie group if its Lie algebra is a simple Lie group. Algebra is.

This is equivalent to the condition that all real normal subgroups are discrete subsets or that there are no nontrivial connected normal subgroups.

For example, SL (2, R) is a simple group in the sense of Lie group theory, but has the normal divisor . The quotient is a simple group, also in the sense of the usual definition in group theory. ${\displaystyle \left\{\pm 1\right\}}$ ${\displaystyle PSL(2,\mathbb {R} )=SL(2,\mathbb {R} )/\left\{\pm 1\right\}}$

See also

• Characteristically simple group

Individual evidence

1. ^ John D. Dixon: Problems in group theory . Dover Publications, Mineola, NY 2007, ISBN 978-0-486-45916-5 , pp. xv .