Free product

In algebra , the free product is a certain construction of a group from two or more given groups. The free product can be thought of as a non- commutative equivalent of the direct sum , much like the equivalent of non-commutative groups to Abelian groups .

construction

If one has given a family of (arbitrary) groups, the free product consists of the set of all finite words (for certain and ), whereby the following conventions should apply: ${\displaystyle \{G_{\alpha }\}_{\alpha \in A}}$ ${\displaystyle {\underset {\alpha \in A}{*}}G_{\alpha }}$ ${\displaystyle g_{\alpha _{1}}\cdots g_{\alpha _{k}}}$ ${\displaystyle \alpha _{i}\in A}$ ${\displaystyle g_{\alpha _{i}}\in G_{\alpha _{i}}}$

• Each element is different from the unit element . ${\displaystyle g_{\alpha }}$ ${\displaystyle G_{\alpha }}$
• ${\displaystyle g_{\alpha _{i}}}$and are not from the same group. ${\displaystyle g_{\alpha _{i+1}}}$

Words that meet these conditions are called reduced. The empty word should also be seen as reduced.

By applying the following two rules, you can always go from any word to a clearly defined reduced word:

• If and are from the same group, then replace the two elements with the product of the two in the group. ${\displaystyle g_{\alpha _{i}}}$ ${\displaystyle g_{\alpha _{i+1}}}$ ${\displaystyle \alpha _{i}=\alpha _{i+1}}$
• If the neutral element is , delete it from the word. ${\displaystyle g_{\alpha }}$ ${\displaystyle G_{\alpha }}$

A group structure can now be defined on the set of reduced words together with the empty word as a unit element. The product is defined by writing one after the other ${\displaystyle {\underset {\alpha \in A}{*}}G_{\alpha }}$

${\displaystyle (g_{\alpha _{1}}\cdots g_{\alpha _{k}})\cdot (h_{\beta _{1}}\cdots h_{\beta _{l}}):=g_{\alpha _{1}}\cdots g_{\alpha _{k}}h_{\beta _{1}}\cdots h_{\beta _{l}}}$

and, if necessary, transition to a reduced word by applying the above rules.

Each group can be viewed as a subgroup in by identifying with the set of words that consist of only one element and the one element . ${\displaystyle G_{\alpha }}$ ${\displaystyle {\underset {\alpha \in A}{*}}G_{\alpha }}$ ${\displaystyle G_{\alpha }}$ ${\displaystyle g\in G_{\alpha }}$

Universal property

The free product of groups fulfills the following universal property : Is

${\displaystyle \{\varphi _{\alpha }:G_{\alpha }\to H\}_{\alpha \in A}}$

a family of homomorphisms , there is exactly one homomorphism , so that for all the identities ${\displaystyle \varphi :{\underset {\alpha \in A}{*}}G_{\alpha }\to H}$ ${\displaystyle \alpha \in A}$

${\displaystyle \varphi \circ i_{\alpha }=\varphi _{\alpha }}$

be valid. It is the identification of described above with the corresponding subgroup in the free product (Compare the corresponding universal property of the direct product : The free product meets exactly the dual universal property and is therefore an example of a co-product ). ${\displaystyle i_{\alpha }:G_{\alpha }\to {\underset {\alpha \in A}{*}}G_{\alpha }}$ ${\displaystyle G_{\alpha }}$

Examples

• If and are topological spaces, and if you consider the one-point union ( wedge ) of the two spaces, that is, if you select one point in each space and "glue" the two spaces together at these two points, then the fundamental group of the result is Space equal to the free product of the fundamental groups of the original spaces: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\vee Y}$

${\displaystyle \pi _{1}(X\vee Y)=\pi _{1}(X)*\pi _{1}(Y)}$.

The Seifert-van Kampen theorem generalizes this principle to associations of spaces that have a complex average (in the aforementioned case is the average of a point).

• The free product of itself, that is , is isomorphic to the free group created by 2 elements . Topologically it results from the above as a fundamental group of a one-point union of two circles, that is, an eight. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} *\mathbb {Z} }$

• More generally, the following applies: The free product of free groups is again a free group, and the powers of the generating systems add up .

• ${\displaystyle \mathbb {Z} _{2}*\mathbb {Z} _{2}\cong D_{\infty }}$. Here is the cyclic group with 2 elements and the infinite dihedral group . ${\displaystyle \mathbb {Z} _{2}}$ ${\displaystyle D_{\infty }}$

• ${\displaystyle \mathbb {Z} _{2}*\mathbb {Z} _{3}\cong \mathrm {PSL} (2,\mathbb {Z} )}$. The right side is the factor group from the special linear group with coefficients from according to their center . ${\displaystyle \mathbb {Z} }$

See also

• Amalgamated product
• Cartesian product
• Direct sum
• Free group
• Coproduct
• Graph theory
• Kurosch subgroup sentence

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 6.2: Free Products of Groups
2. ↑ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 6.2: Examples of Free Products, Example I.
3. Jump up ↑ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 6.2: Examples of Free Products, Example II
4. ↑ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , Chapter 6.2: Examples of Free Products, Example III