Free work

Count Cayley free artwork

\mathbb {Z} _{2}*\mathbb {Z} _{3}

{\ displaystyle \ mathbb {Z} _ {2} * \ mathbb {Z} _ {3}}

{\ displaystyle \ mathbb {Z} _ {2} * \ mathbb {Z} _ {3}}

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A free product of groups is a group generated by elements of these two groups, without any additional relations .

Free work $G_{1}$ ${\ displaystyle G_ {1}}$ $G_ {1}$ and $G_{2}$ ${\ displaystyle G_ {2}}$ $G_ {2}$ usually indicated $G_{1}*G_{2}$ ${\ displaystyle G_ {1} * G_ {2}}$ ${\ displaystyle G_ {1} * G_ {2}}$ .

Definitions

If groups are given through generators and relations $G$ $one$ $=$ $⟨$ $S$ $one$ $|$ $R$ $one$ $⟩$ {\ displaystyle G_ {1} = \ langle S_ {1} | R_ {1} \ rangle} , $G$ $2$ $=$ $⟨$ $S$ $2$ $|$ $R$ $2$ $⟩$ {\ displaystyle G_ {2} = \ langle S_ {2} | R_ {2} \ rangle} then
$G_{1}*G_{2}=\langle S_{1}\cup S_{2}|R_{1}\cup R_{2}\rangle$ ${\ displaystyle G_ {1} * G_ {2} = \ langle S_ {1} \ cup S_ {2} | R_ {1} \ cup R_ {2} \ rangle}$
- This definition also allows a natural generalization to the case of a free product of any number of groups.

Free work $G_{1}*G_{2}$ ${\ displaystyle G_ {1} * G_ {2}}$ can also be defined as layered coproduct $G_{1}\amalg _{\{e\}}G_{2}$ ${\ displaystyle G_ {1} \ amalg _ {\ {e \}} G_ {2}}$ for the trivial group $\{e\}$ ${\ displaystyle \ {e \}}$ in the category of groups.

Examples

Free work $\mathbb {Z} _{2}*\mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2} * \ mathbb {Z} _ {2}}$ isomorphic to the infinite dihedral group $D_{\infty }$ ${\ displaystyle D _ {\ infty}}$ .
Free work $\mathbb {Z} _{2}*\mathbb {Z} _{3}$ ${\ displaystyle \ mathbb {Z} _ {2} * \ mathbb {Z} _ {3}}$ isomorphic to the projective group $PSL(2,\mathbb {Z} )$ ${\ displaystyle PSL (2, \ mathbb {Z})}$ .
Free work $n$ ${\ displaystyle n}$ copies $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ - free group with $n$ ${\ displaystyle n}$ formative.
The Seifert-van Kampen theorem in particular states that if $X$ ${\ displaystyle X}$ - topological space, and $V,U\subset X$ ${\ displaystyle V, U \ subset X}$ - two connected open sets such that the intersection $W=V\cap U$ ${\ displaystyle W = V \ cap U}$ simply connected and $X=V\cup U$ ${\ displaystyle X = V \ cup U}$ then the fundamental group $X$ ${\ displaystyle X}$ there is a free product of fundamental groups $V$ ${\ displaystyle V}$ and $U$ ${\ displaystyle U}$ ; i.e
$\pi _{1}X=\pi _{1}V*\pi _{1}U.$ ${\ displaystyle \ pi _ {1} X = \ pi _ {1} V * \ pi _ {1} U.}$

Literature

Kargapolov M.I., Merzlyakov Yu. I. Fundamentals of group theory. M .: Nauka, 1982.
Kostrikin A. I. Introduction to Algebra. M .: Nauka, 1977.
Kurosh A.G. Group Theory. (3rd ed.). M .: Nauka, 1967.
Hall M. Group Theory. M .: Publishing house of foreign literature, 1962.

Free work

Definitions

Examples

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