Simple group

A simple group is a group that does not have normal subgroups other than the entire group and the unit subgroup.

Finite simple groups are fully classified in 1982.

In the theory of infinite groups, the importance of simple groups is much less because of their immensity.

In the theory of Lie groups and algebraic groups, the definition of a simple group is somewhat different from the above, see a simple Lie group .

Content

1 Examples
- 1.1 Finite simple groups
- 1.2 Infinite simple groups
2 Properties
3 See also

Examples

Finite Simple Groups

Cyclic group $G=\mathbb {Z} /5\mathbb {Z}$ ${\ displaystyle G = \ mathbb {Z} / 5 \ mathbb {Z}}$ is simple. Indeed, if $H$ ${\ displaystyle H}$ - subgroup $G$ ${\ displaystyle G}$ then the order $H$ ${\ displaystyle H}$ by Lagrange's theorem should divide the order $G$ ${\ displaystyle G}$ equal to 5. The only divisors of 5 are 1 or 5, that is, $H$ ${\ displaystyle H}$ either trivial or matches $G$ ${\ displaystyle G}$ . On the contrary, the group $\mathbb {Z} /12\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / 12 \ mathbb {Z}}$ is not simple, since a set consisting of classes of numbers 0, 4, and 8 modulo 12 forms a group of order 3, which is normal as a subgroup of an abelian group. Group $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ integers with the addition operation is also not simple, since the set of even numbers is a nontrivial normal subgroup in $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ . By analogous reasoning, one can verify that all possible simple abelian groups are exactly cyclic groups of simple order.

The classification of simple non-Abelian groups is much more complicated. A simple non-Abelian group of the smallest order - an alternating group $A_{5}$ ${\ displaystyle A_ {5}}$ of order 60, and any simple group of order 60 is isomorphic $A_{5}$ ${\ displaystyle A_ {5}}$ . Moreover, all groups are simple. $A_{n}$ ${\ displaystyle A_ {n}}$ at $n\geqslant 5$ ${\ displaystyle n \ geqslant 5}$ . The next most non-Abelian group after the number of elements $A_{5}$ ${\ displaystyle A_ {5}}$ - special projective group $PSL(2,7)$ ${\ displaystyle PSL (2.7)}$ of order 168. It can be proved that any simple group of order 168 is isomorphic $PSL(2,7)$ ${\ displaystyle PSL (2.7)}$ .

Infinite Simple Groups

Simple is the group of all even permutations, each of which moves a finite subset of elements of an infinite set $X$ ${\ displaystyle X}$ ; in particular, if the set $X$ ${\ displaystyle X}$ countably, this is an infinite alternating group $A_{\infty }$ ${\ displaystyle A _ {\ infty}}$ . Another family example is $PSL_{n}(\mathbb {F} )$ ${\ displaystyle PSL_ {n} (\ mathbb {F})}$ where is the field $\mathbb {F}$ ${\ displaystyle \ mathbb {F}}$ endlessly and $n\geqslant 2$ ${\ displaystyle n \ geqslant 2}$ .

There are finitely generated and even finitely defined infinite simple groups.

Properties

Every group is embeddable in a simple group.