Group Generator

Group Generator $G$ ${\ displaystyle G}$ $G$ (or a set of generators ^[1] , or a system of generators ) is a subset $S$ ${\ displaystyle S}$ $S$ at $G$ ${\ displaystyle G}$ $G$ such that each element $G$ ${\ displaystyle G}$ $G$ can be written as the product of a finite number of elements $S$ ${\ displaystyle S}$ $S$ and their reverse.

Content

Definition

Let be $S$ ${\ displaystyle S}$ - a subset of the group $G$ ${\ displaystyle G}$ . Define $\langle S\rangle$ ${\ displaystyle \ langle S \ rangle}$ , Is the subgroup generated by $S$ ${\ displaystyle S}$ - as the smallest subgroup in $G$ ${\ displaystyle G}$ containing all elements $S$ ${\ displaystyle S}$ , i.e. the intersection of all subgroups containing $S$ ${\ displaystyle S}$ . Equivalently $\langle S\rangle$ ${\ displaystyle \ langle S \ rangle}$ Is a subgroup of all elements $G$ ${\ displaystyle G}$ that can be represented as finite products of elements $S$ ${\ displaystyle S}$ and their reverse .

If a $G=\langle S\rangle$ ${\ displaystyle G = \ langle S \ rangle}$ then they say that $S$ ${\ displaystyle S}$ spawns a group $G$ ${\ displaystyle G}$ . In doing so, the elements $S$ ${\ displaystyle S}$ called group generators . If in a group $G$ ${\ displaystyle G}$ you can choose a finite set of generators, then it is called a finitely generated group .

Remarks

Note that if $S$ ${\ displaystyle S}$ empty then by definition $\langle S\rangle$ ${\ displaystyle \ langle S \ rangle}$ is a trivial group consisting of a neutral element.

When $S$ ${\ displaystyle S}$ contains only one element $x$ ${\ displaystyle x}$ usually write $\langle x\rangle$ ${\ displaystyle \ langle x \ rangle}$ instead $\langle \{x\}\rangle$ ${\ displaystyle \ langle \ {x \} \ rangle}$ . In this case $\langle x\rangle$ ${\ displaystyle \ langle x \ rangle}$ - cyclic subgroup of degrees $x$ ${\ displaystyle x}$ at $G$ ${\ displaystyle G}$ .

Notes

↑ Leng, 1968 , p. 23.

Literature

Leng S. Algebra. - M .: Mir, 1968 .-- 564 p.
Kurosh A.G. Group Theory. - M .: Nauka, 1967 .-- 648 p.

[_f085a1c5b975fda0-1] Leng, 1968 , p. 23.